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Theorem opelvv 5035
Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opelvv.1  |-  A  e. 
_V
opelvv.2  |-  B  e. 
_V
Assertion
Ref Expression
opelvv  |-  <. A ,  B >.  e.  ( _V 
X.  _V )

Proof of Theorem opelvv
StepHypRef Expression
1 opelvv.1 . 2  |-  A  e. 
_V
2 opelvv.2 . 2  |-  B  e. 
_V
3 opelxpi 5020 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )
41, 2, 3mp2an 670 1  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1823   _Vcvv 3106   <.cop 4022    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498  df-xp 4994
This theorem is referenced by:  relsnop  5095  relopabi  5116  1st2ndb  6811  eqop2  6814  evlfcl  15690  brtxp  29758  brpprod  29763  brsset  29767  brcart  29810  brcup  29817  brcap  29818  fusgraimpcl  32799  fusgraimpclALT  32801
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