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Theorem opth2 4250
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 4249 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
41, 2, 3mp2an 653 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   _Vcvv 2790   <.cop 3645
This theorem is referenced by:  eqvinop  4253  opelxp  4721  fsn  5698  opiota  6292  canthwe  8275  ltresr  8764  diblsmopel  31434  cdlemn7  31466  dihordlem7  31477  xihopellsmN  31517  dihopellsm  31518  dihpN  31599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651
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