Theorem opprc1 4226

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4225. (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 455	. . 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 4225	. 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 367   = wceq 1398   e. wcel 1823   _Vcvv 3106   (/)c0 3783   <.cop 4022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-op 4023

This theorem is referenced by:  brprcneu  5841  eu2ndop1stv  32446  bj-inftyexpidisj  35013