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Theorem altopth2 25527
Description: Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopth2  |-  ( B  e.  V  ->  ( << A ,  B >>  = 
<< C ,  D >>  ->  B  =  D )
)

Proof of Theorem altopth2
StepHypRef Expression
1 altopthsn 25522 . 2  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } ) )
2 sneqrg 3912 . . 3  |-  ( B  e.  V  ->  ( { B }  =  { D }  ->  B  =  D ) )
32adantld 454 . 2  |-  ( B  e.  V  ->  (
( { A }  =  { C }  /\  { B }  =  { D } )  ->  B  =  D ) )
41, 3syl5bi 209 1  |-  ( B  e.  V  ->  ( << A ,  B >>  = 
<< C ,  D >>  ->  B  =  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3759   <<caltop 25517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-sn 3765  df-pr 3766  df-altop 25519
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