Theorem opprc 4424

Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

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

Proof of Theorem opprc

Step	Hyp	Ref	Expression
1		dfopif 4399	. 2 |- <. A , B >. = if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) )
2		iffalse 4095	. 2 |- ( -. ( A e. _V /\ B e. _V ) -> if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) ) = (/) )
3	1, 2	syl5eq 2668	1 |- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-op 4184

This theorem is referenced by:  opprc1  4425  opprc2  4426  oprcl  4427  opnz  4942  opswap  5622  brabv  6699  brtpos  7361  elnonrel  37891