Theorem opelvv 4805

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 4786	. 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 2205   _Vcvv 2815   <.cop 3697   X. cxp 4752

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177  df-xp 4760

This theorem is referenced by:  relsnop  4861  relopabi  4885  eqop2  6385