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

Theorem opelvv 5046
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 5031 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  e.  ( _V  X.  _V ) )
41, 2, 3mp2an 672 1  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   _Vcvv 3113   <.cop 4033    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005
This theorem is referenced by:  relsnop  5107  relopabi  5128  1st2ndb  6823  eqop2  6826  evlfcl  15352  brtxp  29383  brpprod  29388  brsset  29392  brcart  29435  brcup  29442  brcap  29443  fusgraimpcl  32121  fusgraimpclALT  32123
  Copyright terms: Public domain W3C validator