Theorem brvvdif 34027

Description: Binary relation with the complement under the universal class of ordered pairs. (Contributed by Peter Mazsa, 9-Nov-2018.)

Assertion

Ref	Expression
brvvdif	|- ( ( A e. V /\ B e. W ) -> ( A ( ( _V X. _V ) \ R ) B <-> -. A R B ) )

Proof of Theorem brvvdif

Step	Hyp	Ref	Expression
1		opelvvdif 34023	. 2 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. ( ( _V X. _V ) \ R ) <-> -. <. A , B >. e. R ) )
2		df-br 4654	. 2 |- ( A ( ( _V X. _V ) \ R ) B <-> <. A , B >. e. ( ( _V X. _V ) \ R ) )
3		df-br 4654	. . 3 |- ( A R B <-> <. A , B >. e. R )
4	3	notbii 310	. 2 |- ( -. A R B <-> -. <. A , B >. e. R )
5	1, 2, 4	3bitr4g 303	1 |- ( ( A e. V /\ B e. W ) -> ( A ( ( _V X. _V ) \ R ) B <-> -. A R B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   \ cdif 3571   <.cop 4183   class class class wbr 4653   X. cxp 5112

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120

This theorem is referenced by:  brvbrvvdif  34028