Theorem opprc2 3797

Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 3795. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opprc2	|- ( -. B e. _V -> <. A , B >. = (/) )

Proof of Theorem opprc2

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 |- ( ( A e. _V /\ B e. _V ) -> B e. _V )
2	1	con3i 632	. 2 |- ( -. B e. _V -> -. ( A e. _V /\ B e. _V ) )
3		opprc 3795	. 2 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
4	2, 3	syl 14	1 |- ( -. B e. _V -> <. A , B >. = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   _Vcvv 2735   (/)c0 3420   <.cop 3592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-dif 3129  df-nul 3421  df-op 3598

This theorem is referenced by: (None)