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Theorem opth2 4430
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 4429 . 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 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809
This theorem is referenced by:  eqvinop  4433  opelxp  4900  fsn  5898  opiota  6527  canthwe  8518  ltresr  9007  fmucndlem  18313  diblsmopel  31906  cdlemn7  31938  dihordlem7  31949  xihopellsmN  31989  dihopellsm  31990  dihpN  32071
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815
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