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Theorem zfpair2 5294
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 5293. See zfpair 5285 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 5293 . . . 4 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
21bm1.3ii 5167 . . 3 𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦))
3 dfcleq 2731 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
4 vex 3401 . . . . . . . 8 𝑤 ∈ V
54elpr 4536 . . . . . . 7 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
65bibi2i 341 . . . . . 6 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
76albii 1826 . . . . 5 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
83, 7bitri 278 . . . 4 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
98exbii 1854 . . 3 (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
102, 9mpbir 234 . 2 𝑧 𝑧 = {𝑥, 𝑦}
1110issetri 3413 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wb 209  wo 846  wal 1540   = wceq 1542  wex 1786  wcel 2113  Vcvv 3397  {cpr 4515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710  ax-sep 5164  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3399  df-un 3846  df-sn 4514  df-pr 4516
This theorem is referenced by:  snex  5295  prex  5296  pwssun  5421  xpsspw  5647  funopg  6367  fiint  8862  brdom7disj  10024  brdom6disj  10025  2pthfrgrrn  28211  sprval  44449  prprval  44484  reupr  44492
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