MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelvvg Structured version   Unicode version

Theorem opelvvg 4923
Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
opelvvg  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )

Proof of Theorem opelvvg
StepHypRef Expression
1 elex 2964 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2964 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 opelxpi 4910 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )
41, 2, 3syl2an 464 1  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   _Vcvv 2956   <.cop 3817    X. cxp 4876
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884
  Copyright terms: Public domain W3C validator