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Theorem axpr 4935
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 4936 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 4934 . . 3 {𝑥, 𝑦} ∈ V
21isseti 3240 . 2 𝑧 𝑧 = {𝑥, 𝑦}
3 dfcleq 2645 . . 3 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
4 vex 3234 . . . . . . 7 𝑤 ∈ V
54elpr 4231 . . . . . 6 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
65bibi2i 326 . . . . 5 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
7 biimpr 210 . . . . 5 ((𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)) → ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
86, 7sylbi 207 . . . 4 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) → ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
98alimi 1779 . . 3 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
103, 9sylbi 207 . 2 (𝑧 = {𝑥, 𝑦} → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
112, 10eximii 1804 1 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213
This theorem is referenced by: (None)
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