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Theorem zfpair2 5321
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 5320. See zfpair 5312 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
Assertion
Ref Expression
zfpair2 {𝑥, 𝑦} ∈ V

Proof of Theorem zfpair2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-pr 5320 . . . 4 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
21bm1.3ii 5197 . . 3 𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦))
3 dfcleq 2813 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
4 vex 3496 . . . . . . . 8 𝑤 ∈ V
54elpr 4582 . . . . . . 7 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
65bibi2i 340 . . . . . 6 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
76albii 1813 . . . . 5 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
83, 7bitri 277 . . . 4 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
98exbii 1841 . . 3 (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
102, 9mpbir 233 . 2 𝑧 𝑧 = {𝑥, 𝑦}
1110issetri 3509 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wb 208  wo 843  wal 1528   = wceq 1530  wex 1773  wcel 2107  Vcvv 3493  {cpr 4561
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-un 3939  df-sn 4560  df-pr 4562
This theorem is referenced by:  snex  5322  prex  5323  pwssun  5448  xpsspw  5675  funopg  6382  fiint  8787  brdom7disj  9945  brdom6disj  9946  2pthfrgrrn  28053  sprval  43626  prprval  43661  reupr  43669
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