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

Theorem axpr 4679
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 4680 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 14-Nov-2006.) (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
StepHypRef Expression
1 zfpair 4678 . . 3 {𝑥, 𝑦} ∈ V
21isseti 3072 . 2 𝑧 𝑧 = {𝑥, 𝑦}
3 dfcleq 2499 . . 3 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
4 vex 3069 . . . . . . 7 𝑤 ∈ V
54elpr 4016 . . . . . 6 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
65bibi2i 322 . . . . 5 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
7 biimpr 205 . . . . 5 ((𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)) → ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
86, 7sylbi 202 . . . 4 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) → ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
98alimi 1713 . . 3 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
103, 9sylbi 202 . 2 (𝑧 = {𝑥, 𝑦} → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
112, 10eximii 1740 1 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  wo 377  wal 1466   = wceq 1468  wex 1692  wcel 1937  {cpr 3997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-rep 4548  ax-sep 4558  ax-nul 4567  ax-pow 4619
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-v 3068  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3758  df-pw 3980  df-sn 3996  df-pr 3998
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator