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Mirrors > Home > MPE Home > Th. List > dfcleq  Structured version GIF version 
Description: Define the equality
connective between classes. Definition 2.7 of
[Quine] p. 18. Also Definition 4.5 of
[TakeutiZaring] p. 13; Chapter 4
provides its justification and methods for eliminating it. Note that
its elimination will not necessarily result in a single wff in the
original language but possibly a "scheme" of wffs.
This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce 𝑦 = 𝑧 ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2427). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated. We could avoid this complication by introducing a new symbol, say =_{2}, in place of =. This would also have the advantage of making elimination of the definition straightforward, so that we could eliminate Extensionality as a hypothesis. We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play rather than just being an artifact of a definition. One of our theorems would then be 𝑥 =_{2} 𝑦 ↔ 𝑥 = 𝑦 by invoking Extensionality. However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality. See also comments under dfclab 2430, dfclel 2439, and abeq2 2548. In the form of dfcleq 2437, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 15Sep1993.) 
Ref  Expression 

dfcleq.1  ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) 
Ref  Expression 

dfcleq  ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) 
Step  Hyp  Ref  Expression 

1  cA  . . 3 class 𝐴  
2  cB  . . 3 class 𝐵  
3  1, 2  wceq 1654  . 2 wff 𝐴 = 𝐵 
4  vx  . . . . . 6 set 𝑥  
5  4  cv 1653  . . . . 5 class 𝑥 
6  5, 1  wcel 1728  . . . 4 wff 𝑥 ∈ 𝐴 
7  5, 2  wcel 1728  . . . 4 wff 𝑥 ∈ 𝐵 
8  6, 7  wb 178  . . 3 wff (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) 
9  8, 4  wal 1550  . 2 wff ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) 
10  3, 9  wb 178  1 wff (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) 
Colors of variables: wff set class 
This definition is referenced by: dfcleq 2437 
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