ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prid1 Unicode version

Theorem prid1 3695
Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
prid1.1  |-  A  e. 
_V
Assertion
Ref Expression
prid1  |-  A  e. 
{ A ,  B }

Proof of Theorem prid1
StepHypRef Expression
1 prid1.1 . 2  |-  A  e. 
_V
2 prid1g 3693 . 2  |-  ( A  e.  _V  ->  A  e.  { A ,  B } )
31, 2ax-mp 5 1  |-  A  e. 
{ A ,  B }
Colors of variables: wff set class
Syntax hints:    e. wcel 2146   _Vcvv 2735   {cpr 3590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596
This theorem is referenced by:  prid2  3696  prnz  3711  preqr1  3764  preq12b  3766  prel12  3767  opi1  4226  opeluu  4444  onsucelsucexmidlem1  4521  regexmidlem1  4526  reg2exmidlema  4527  opthreg  4549  ordtri2or2exmid  4564  ontri2orexmidim  4565  dmrnssfld  4883  funopg  5242  acexmidlemb  5857  0lt2o  6432  2dom  6795  unfiexmid  6907  djuss  7059  exmidomni  7130  exmidonfinlem  7182  exmidaclem  7197  reelprrecn  7921  pnfxr  7984  sup3exmid  8885  lgsdir2lem3  13924  bdop  14109  2o01f  14228  iswomni0  14282
  Copyright terms: Public domain W3C validator