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Theorem opth2 4185
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 4184 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
41, 2, 3mp2an 656 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2740   <.cop 3584
This theorem is referenced by:  eqvinop  4188  opelxp  4672  fsn  5595  opiota  6221  canthwe  8206  ltresr  8695  diblsmopel  30491  cdlemn7  30523  dihordlem7  30534  xihopellsmN  30574  dihopellsm  30575  dihpN  30656
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590
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