Theorem opprc1 3629

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 3628. (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 107	. . 3 |- ( ( A e. _V /\ B e. _V ) -> A e. _V )
2	1	con3i 595	. 2 |- ( -. A e. _V -> -. ( A e. _V /\ B e. _V ) )
3		opprc 3628	. 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 102   = wceq 1287   e. wcel 1436   _Vcvv 2615   (/)c0 3275   <.cop 3434

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990  df-nul 3276  df-op 3440

This theorem is referenced by:  brprcneu  5263