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Theorem zfpair2 5428
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 5427. See zfpair 5419 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 5427 . . . 4 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
21bm1.3ii 5302 . . 3 𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦))
3 dfcleq 2724 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
4 vex 3477 . . . . . . . 8 𝑤 ∈ V
54elpr 4651 . . . . . . 7 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
65bibi2i 337 . . . . . 6 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
76albii 1820 . . . . 5 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
83, 7bitri 275 . . . 4 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
98exbii 1849 . . 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 844  wal 1538   = wceq 1540  wex 1780  wcel 2105  Vcvv 3473  {cpr 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-sn 4629  df-pr 4631
This theorem is referenced by:  vsnex  5429  prex  5432  pwssun  5571  xpsspw  5809  funopg  6582  fiint  9330  brdom7disj  10532  brdom6disj  10533  2pthfrgrrn  29817  sprval  46458  prprval  46493  reupr  46501
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