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Theorem axpr 5316
Description: Unabbreviated version of the Axiom of Pairing of ZF set theory, derived as a theorem from the other axioms.

This theorem should not be referenced by any proof. Instead, use ax-pr 5317 below so that the uses of the Axiom of Pairing can be more easily identified.

For a shorter proof using ax-ext 2796, see axprALT 5310. (Contributed by NM, 14-Nov-2006.) Remove dependency on ax-ext 2796. (Revised by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by BJ, 13-Aug-2023.) Use ax-pr 5317 instead. (New usage is discouraged.)

Assertion
Ref Expression
axpr 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤

Proof of Theorem axpr
Dummy variables 𝑠 𝑝 𝑡 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axprlem3 5313 . . . 4 𝑧𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
2 biimpr 223 . . . . 5 ((𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) → (∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧))
32alimi 1813 . . . 4 (∀𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) → ∀𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧))
41, 3eximii 1838 . . 3 𝑧𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧)
5 axprlem4 5314 . . . . . . . 8 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
6 axprlem5 5315 . . . . . . . 8 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
75, 6jaodan 955 . . . . . . 7 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ (𝑤 = 𝑥𝑤 = 𝑦)) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
87ex 416 . . . . . 6 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → ((𝑤 = 𝑥𝑤 = 𝑦) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
98imim1d 82 . . . . 5 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → ((∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧) → ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
109alimdv 1918 . . . 4 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (∀𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧) → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
1110eximdv 1919 . . 3 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (∃𝑧𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧) → ∃𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
124, 11mpi 20 . 2 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → ∃𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
13 axprlem2 5312 . 2 𝑝𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝)
1412, 13exlimiiv 1933 1 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  if-wif 1058  wal 1536  wex 1781  wral 3133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-tru 1541  df-ex 1782  df-nf 1786  df-ral 3138
This theorem is referenced by: (None)
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