Theorem opprc1 3727

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 3726. (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 621	. 2 |- ( -. A e. _V -> -. ( A e. _V /\ B e. _V ) )
3		opprc 3726	. 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 1331   e. wcel 1480   _Vcvv 2686   (/)c0 3363   <.cop 3530

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364  df-op 3536

This theorem is referenced by:  brprcneu  5414