MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prid2g Unicode version

Theorem prid2g 3707
Description: An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
Assertion
Ref Expression
prid2g  |-  ( B  e.  V  ->  B  e.  { A ,  B } )

Proof of Theorem prid2g
StepHypRef Expression
1 prid1g 3706 . 2  |-  ( B  e.  V  ->  B  e.  { B ,  A } )
2 prcom 3679 . 2  |-  { B ,  A }  =  { A ,  B }
31, 2syl6eleq 2348 1  |-  ( B  e.  V  ->  B  e.  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   {cpr 3615
This theorem is referenced by:  fr2nr  4343  unisn2  4494  pw2f1olem  6934  gcdcllem3  12654  indistopon  16700  pptbas  16707  coseq0negpitopi  19833  vdgr1b  23267  fnckle  25412  pmtrprfv  26763
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 2239
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 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-un 3132  df-sn 3620  df-pr 3621
  Copyright terms: Public domain W3C validator