Theorem opprc1 3787

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 3786. (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 108	. . 3 |- ( ( A e. _V /\ B e. _V ) -> A e. _V )
2	1	con3i 627	. 2 |- ( -. A e. _V -> -. ( A e. _V /\ B e. _V ) )
3		opprc 3786	. 2 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
4	2, 3	syl 14	1 |- ( -. A e. _V -> <. A , B >. = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   (/)c0 3414   <.cop 3586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415  df-op 3592

This theorem is referenced by:  brprcneu  5489