Theorem opprc1 4183

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4182. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opprc1	|- ( -. A e. _V -> <. A , B >. = (/) )

Proof of Theorem opprc1

Step	Hyp	Ref	Expression
1		simpl 457	. . 3 |- ( ( A e. _V /\ B e. _V ) -> A e. _V )
2	1	con3i 135	. 2 |- ( -. A e. _V -> -. ( A e. _V /\ B e. _V ) )
3		opprc 4182	. 2 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
4	2, 3	syl 16	1 |- ( -. A e. _V -> <. A , B >. = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   _Vcvv 3071   (/)c0 3738   <.cop 3984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3073  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-op 3985

This theorem is referenced by:  brprcneu  5785  eu2ndop1stv  30167  bj-inftyexpidisj  32842