Theorem opprc2 3885

Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 3883. (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 637	. 2 |- ( -. B e. _V -> -. ( A e. _V /\ B e. _V ) )
3		opprc 3883	. 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 1397   e. wcel 2202   _Vcvv 2802   (/)c0 3494   <.cop 3672

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-nul 3495  df-op 3678

This theorem is referenced by: (None)