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Theorem opth2 4379
Description: Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
Hypotheses
Ref Expression
opth2.1  |-  C  e. 
_V
opth2.2  |-  D  e. 
_V
Assertion
Ref Expression
opth2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)

Proof of Theorem opth2
StepHypRef Expression
1 opth2.1 . 2  |-  C  e. 
_V
2 opth2.2 . 2  |-  D  e. 
_V
3 opthg2 4378 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
41, 2, 3mp2an 654 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   <.cop 3760
This theorem is referenced by:  eqvinop  4382  opelxp  4848  fsn  5845  opiota  6471  canthwe  8459  ltresr  8948  fmucndlem  18242  diblsmopel  31286  cdlemn7  31318  dihordlem7  31329  xihopellsmN  31369  dihopellsm  31370  dihpN  31451
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 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766
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