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Theorem 2eu5 2385
Description: An alternate definition of double existential uniqueness (see 2eu4 2384). A mistake sometimes made in the literature is to use ∃!𝑥∃!𝑦 to mean "exactly one 𝑥 and exactly one 𝑦." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining 𝑥∃*𝑦𝜑 as an additional condition. The correct definition apparently has never been published. (∃* means "exists at most one.") (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
2eu5 ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2eu5
StepHypRef Expression
1 2eu1 2381 . . 3 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
21pm5.32ri 621 . 2 ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ∧ ∀𝑥∃*𝑦𝜑))
3 eumo 2341 . . . . 5 (∃!𝑦𝑥𝜑 → ∃*𝑦𝑥𝜑)
43adantl 454 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃*𝑦𝑥𝜑)
5 2moex 2372 . . . 4 (∃*𝑦𝑥𝜑 → ∀𝑥∃*𝑦𝜑)
64, 5syl 16 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∀𝑥∃*𝑦𝜑)
76pm4.71i 615 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ∧ ∀𝑥∃*𝑦𝜑))
8 2eu4 2384 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
92, 7, 83bitr2i 266 1 ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 178  wa 360  wal 1551  wex 1552  ∃!weu 2301  ∃*wmo 2302
This theorem is referenced by:  2reu5lem3  3163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1557  ax-5 1568  ax-17 1629  ax-9 1670  ax-8 1691  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1954
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1330  df-ex 1553  df-nf 1556  df-sb 1662  df-eu 2305  df-mo 2306
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