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Theorem axpr 5427
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 5432 below so that the uses of the Axiom of Pairing can be more easily identified.

For a shorter proof using ax-ext 2708, see axprALT 5422. (Contributed by NM, 14-Nov-2006.) Remove dependency on ax-ext 2708. (Revised by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by BJ, 13-Aug-2023.) (Proof shortened by Matthew House, 18-Sep-2025.) Use ax-pr 5432 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 5425 . . 3 𝑧𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
2 axprlem1 5423 . . . . . . . . 9 𝑠𝑛(∀𝑡 ¬ 𝑡𝑛𝑛𝑠)
32sepexi 5301 . . . . . . . 8 𝑠𝑛(𝑛𝑠 ↔ ∀𝑡 ¬ 𝑡𝑛)
4 biimp 215 . . . . . . . 8 ((𝑛𝑠 ↔ ∀𝑡 ¬ 𝑡𝑛) → (𝑛𝑠 → ∀𝑡 ¬ 𝑡𝑛))
5 ax-nul 5306 . . . . . . . . . 10 𝑛𝑡 ¬ 𝑡𝑛
6 exbi 1847 . . . . . . . . . 10 (∀𝑛(𝑛𝑠 ↔ ∀𝑡 ¬ 𝑡𝑛) → (∃𝑛 𝑛𝑠 ↔ ∃𝑛𝑡 ¬ 𝑡𝑛))
75, 6mpbiri 258 . . . . . . . . 9 (∀𝑛(𝑛𝑠 ↔ ∀𝑡 ¬ 𝑡𝑛) → ∃𝑛 𝑛𝑠)
8 ifptru 1075 . . . . . . . . 9 (∃𝑛 𝑛𝑠 → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑥))
97, 8syl 17 . . . . . . . 8 (∀𝑛(𝑛𝑠 ↔ ∀𝑡 ¬ 𝑡𝑛) → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑥))
103, 4, 9axprlem4 5426 . . . . . . 7 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (𝑤 = 𝑥 → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
11 ax-nul 5306 . . . . . . . 8 𝑠𝑛 ¬ 𝑛𝑠
12 pm2.21 123 . . . . . . . 8 𝑛𝑠 → (𝑛𝑠 → ∀𝑡 ¬ 𝑡𝑛))
13 alnex 1781 . . . . . . . . 9 (∀𝑛 ¬ 𝑛𝑠 ↔ ¬ ∃𝑛 𝑛𝑠)
14 ifpfal 1076 . . . . . . . . 9 (¬ ∃𝑛 𝑛𝑠 → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑦))
1513, 14sylbi 217 . . . . . . . 8 (∀𝑛 ¬ 𝑛𝑠 → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑦))
1611, 12, 15axprlem4 5426 . . . . . . 7 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (𝑤 = 𝑦 → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
1710, 16jaod 860 . . . . . 6 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → ((𝑤 = 𝑥𝑤 = 𝑦) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
18 imbi2 348 . . . . . 6 ((𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) → (((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧) ↔ ((𝑤 = 𝑥𝑤 = 𝑦) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))))
1917, 18syl5ibrcom 247 . . . . 5 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → ((𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) → ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
2019alimdv 1916 . . . 4 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (∀𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
2120eximdv 1917 . . 3 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (∃𝑧𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) → ∃𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
221, 21mpi 20 . 2 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → ∃𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
23 axprlem2 5424 . 2 𝑝𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝)
2422, 23exlimiiv 1931 1 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  if-wif 1063  wal 1538  wex 1779  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-ex 1780  df-ral 3062
This theorem is referenced by: (None)
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