Theorem opelvv 4743

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

Step	Hyp	Ref	Expression
1		opelvv.1	. 2 |- A e. _V
2		opelvv.2	. 2 |- B e. _V
3		opelxpi 4725	. 2 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. e. ( _V X. _V ) )
4	1, 2, 3	mp2an 426	1 |- <. A , B >. e. ( _V X. _V )

Syntax hints:   e. wcel 2178   _Vcvv 2776   <.cop 3646   X. cxp 4691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4122  df-xp 4699

This theorem is referenced by:  relsnop  4799  relopabi  4821  eqop2  6287