Theorem opelvv 4517

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 4499	. 2 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. e. ( _V X. _V ) )
4	1, 2, 3	mp2an 418	1 |- <. A , B >. e. ( _V X. _V )

Syntax hints:   e. wcel 1445   _Vcvv 2633   <.cop 3469   X. cxp 4465

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-opab 3922  df-xp 4473

This theorem is referenced by:  relsnop  4573  relopabi  4593  eqop2  5986