Theorem brnonrel 37895

Description: A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.)

Assertion

Ref	Expression
brnonrel	|- ( ( X e. U /\ Y e. V ) -> -. X ( A \ `' `' A ) Y )

Proof of Theorem brnonrel

Step	Hyp	Ref	Expression
1		br0 4701	. 2 |- -. Y (/) X
2		cnvnonrel 37894	. . . 4 |- `' ( A \ `' `' A ) = (/)
3	2	breqi 4659	. . 3 |- ( Y `' ( A \ `' `' A ) X <-> Y (/) X )
4		brcnvg 5303	. . . 4 |- ( ( Y e. V /\ X e. U ) -> ( Y `' ( A \ `' `' A ) X <-> X ( A \ `' `' A ) Y ) )
5	4	ancoms 469	. . 3 |- ( ( X e. U /\ Y e. V ) -> ( Y `' ( A \ `' `' A ) X <-> X ( A \ `' `' A ) Y ) )
6	3, 5	syl5rbbr 275	. 2 |- ( ( X e. U /\ Y e. V ) -> ( X ( A \ `' `' A ) Y <-> Y (/) X ) )
7	1, 6	mtbiri 317	1 |- ( ( X e. U /\ Y e. V ) -> -. X ( A \ `' `' A ) Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   \ cdif 3571   (/)c0 3915   class class class wbr 4653   `'ccnv 5113

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  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  df-rel 5121  df-cnv 5122

This theorem is referenced by: (None)