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Theorem altopth1 25810
Description: Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopth1  |-  ( A  e.  V  ->  ( << A ,  B >>  = 
<< C ,  D >>  ->  A  =  C )
)

Proof of Theorem altopth1
StepHypRef Expression
1 altopthsn 25806 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } ) )
2 sneqrg 3968 . . 3  |-  ( A  e.  V  ->  ( { A }  =  { C }  ->  A  =  C ) )
32adantrd 455 . 2  |-  ( A  e.  V  ->  (
( { A }  =  { C }  /\  { B }  =  { D } )  ->  A  =  C ) )
41, 3syl5bi 209 1  |-  ( A  e.  V  ->  ( << A ,  B >>  = 
<< C ,  D >>  ->  A  =  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3814   <<caltop 25801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820  df-pr 3821  df-altop 25803
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