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

Theorem prid1g 3692
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 2256 . . 3  |-  A  =  A
21orci 381 . 2  |-  ( A  =  A  \/  A  =  B )
3 elprg 3617 . 2  |-  ( A  e.  V  ->  ( A  e.  { A ,  B }  <->  ( A  =  A  \/  A  =  B ) ) )
42, 3mpbiri 226 1  |-  ( A  e.  V  ->  A  e.  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    = wceq 1619    e. wcel 1621   {cpr 3601
This theorem is referenced by:  prid2g  3693  prid1  3694  opth1  4202  fr2nr  4329  fveqf1o  5726  pw2f1olem  6920  gcdcllem3  12640  pptbas  16693  coseq0negpitopi  19819  vdgr1b  23253  fnckle  25398  kelac2  26516  pmtrprfv  26749
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-un 3118  df-sn 3606  df-pr 3607
  Copyright terms: Public domain W3C validator