HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version
Theorem zfpair2 2770
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 2769. See zfpair 2767 for its derivation from the other axioms.
Assertion
Ref Expression
zfpair2 |- {x, y} e. V
Proof of Theorem zfpair2
StepHypRef Expression
1 ax-pr 2769 . . . 4 |- E.zA.w((w = x \/ w = y) -> w e. z)
21bm1.3ii 2696 . . 3 |- E.zA.w(w e. z <-> (w = x \/ w = y))
3 dfcleq 1463 . . . . 5 |- (z = {x, y} <-> A.w(w e. z <-> w e. {x, y}))
4 visset 1804 . . . . . . . 8 |- w e. V
54elpr 2414 . . . . . . 7 |- (w e. {x, y} <-> (w = x \/ w = y))
65bibi2i 606 . . . . . 6 |- ((w e. z <-> w e. {x, y}) <-> (w e. z <-> (w = x \/ w = y)))
76albii 996 . . . . 5 |- (A.w(w e. z <-> w e. {x, y}) <-> A.w(w e. z <-> (w = x \/ w = y)))
83, 7bitr 173 . . . 4 |- (z = {x, y} <-> A.w(w e. z <-> (w = x \/ w = y)))
98exbii 1047 . . 3 |- (E.z z = {x, y} <-> E.zA.w(w e. z <-> (w = x \/ w = y)))
102, 9mpbir 190 . 2 |- E.z z = {x, y}
1110issetri 1807 1 |- {x, y} e. V
Colors of variables: wff set class
Syntax hints:   <-> wb 146   \/ wo 222  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802  {cpr 2400
This theorem is referenced by:  prex 2771  pwssun 2816  fr2nr 2915  xpsspw 3247  funopg 3533  fiint 4534  brdom7disj 4776  brdom6disj 4777
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pr 2769
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403
Copyright terms: Public domain