Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  altopthg Unicode version

Theorem altopthg 24501
Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopthg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )

Proof of Theorem altopthg
StepHypRef Expression
1 altopthsn 24495 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } ) )
2 sneqbg 3783 . . 3  |-  ( A  e.  V  ->  ( { A }  =  { C }  <->  A  =  C
) )
3 sneqbg 3783 . . 3  |-  ( B  e.  W  ->  ( { B }  =  { D }  <->  B  =  D
) )
42, 3bi2anan9 843 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( { A }  =  { C }  /\  { B }  =  { D } )  <-> 
( A  =  C  /\  B  =  D ) ) )
51, 4syl5bb 248 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640   <<caltop 24490
This theorem is referenced by:  altopth  24503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-altop 24492
  Copyright terms: Public domain W3C validator