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Theorem opelvv 4596
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 4578 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )
41, 2, 3mp2an 423 1  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
Colors of variables: wff set class
Syntax hints:    e. wcel 1481   _Vcvv 2689   <.cop 3534    X. cxp 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-opab 3997  df-xp 4552
This theorem is referenced by:  relsnop  4652  relopabi  4672  eqop2  6083
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