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Theorem zfpair2 5434
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 5433. See zfpair 5425 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 5433 . . . 4 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
21bm1.3ii 5306 . . 3 𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦))
3 dfcleq 2721 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
4 vex 3477 . . . . . . . 8 𝑤 ∈ V
54elpr 4656 . . . . . . 7 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
65bibi2i 336 . . . . . 6 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
76albii 1813 . . . . 5 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
83, 7bitri 274 . . . 4 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
98exbii 1842 . . 3 (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
102, 9mpbir 230 . 2 𝑧 𝑧 = {𝑥, 𝑦}
1110issetri 3490 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 845  wal 1531   = wceq 1533  wex 1773  wcel 2098  Vcvv 3473  {cpr 4634
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-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-un 3954  df-sn 4633  df-pr 4635
This theorem is referenced by:  vsnex  5435  prex  5438  pwssun  5577  xpsspw  5815  funopg  6592  fiint  9356  brdom7disj  10562  brdom6disj  10563  2pthfrgrrn  30112  sprval  46848  prprval  46883  reupr  46891  uspgrimprop  47249
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