MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axpr Structured version   Visualization version   GIF version

Theorem axpr 5430
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 5431 below so that the uses of the Axiom of Pairing can be more easily identified.

For a shorter proof using ax-ext 2698, see axprALT 5424. (Contributed by NM, 14-Nov-2006.) Remove dependency on ax-ext 2698. (Revised by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by BJ, 13-Aug-2023.) Use ax-pr 5431 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 5427 . . . 4 𝑧𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
2 biimpr 219 . . . . 5 ((𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) → (∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧))
32alimi 1805 . . . 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 1831 . . 3 𝑧𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧)
5 axprlem4 5428 . . . . . . . 8 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
6 axprlem5 5429 . . . . . . . 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 411 . . . . . 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 1911 . . . 4 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (∀𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧) → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
1110eximdv 1912 . . 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 5426 . 2 𝑝𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝)
1412, 13exlimiiv 1926 1 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  if-wif 1060  wal 1531  wex 1773  wral 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-tru 1536  df-ex 1774  df-nf 1778  df-ral 3058
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator