MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prid1g Unicode version

Theorem prid1g 3745
Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
Assertion
Ref Expression
prid1g  |-  ( A  e.  V  ->  A  e.  { A ,  B } )

Proof of Theorem prid1g
StepHypRef Expression
1 eqid 2296 . . 3  |-  A  =  A
21orci 379 . 2  |-  ( A  =  A  \/  A  =  B )
3 elprg 3670 . 2  |-  ( A  e.  V  ->  ( A  e.  { A ,  B }  <->  ( A  =  A  \/  A  =  B ) ) )
42, 3mpbiri 224 1  |-  ( A  e.  V  ->  A  e.  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1632    e. wcel 1696   {cpr 3654
This theorem is referenced by:  prid2g  3746  prid1  3747  opth1  4260  fr2nr  4387  fveqf1o  5822  pw2f1olem  6982  gcdcllem3  12708  pptbas  16761  coseq0negpitopi  19887  vdgr1b  23910  fnckle  26148  kelac2  27266  pmtrprfv  27499  nb3graprlem1  28287  nb3graprlem2  28288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-pr 3660
  Copyright terms: Public domain W3C validator