Theorem cononrel2 37901

Description: Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)

Assertion

Ref	Expression
cononrel2	|- ( A o. ( B \ `' `' B ) ) = (/)

Proof of Theorem cononrel2

Step	Hyp	Ref	Expression
1		cnvco 5308	. . . 4 |- `' ( A o. ( B \ `' `' B ) ) = ( `' ( B \ `' `' B ) o. `' A )
2		cnvnonrel 37894	. . . . 5 |- `' ( B \ `' `' B ) = (/)
3	2	coeq1i 5281	. . . 4 |- ( `' ( B \ `' `' B ) o. `' A ) = ( (/) o. `' A )
4		co01 5650	. . . 4 |- ( (/) o. `' A ) = (/)
5	1, 3, 4	3eqtri 2648	. . 3 |- `' ( A o. ( B \ `' `' B ) ) = (/)
6	5	cnveqi 5297	. 2 |- `' `' ( A o. ( B \ `' `' B ) ) = `' (/)
7		relco 5633	. . 3 |- Rel ( A o. ( B \ `' `' B ) )
8		dfrel2 5583	. . 3 |- ( Rel ( A o. ( B \ `' `' B ) ) <-> `' `' ( A o. ( B \ `' `' B ) ) = ( A o. ( B \ `' `' B ) ) )
9	7, 8	mpbi 220	. 2 |- `' `' ( A o. ( B \ `' `' B ) ) = ( A o. ( B \ `' `' B ) )
10		cnv0 5535	. 2 |- `' (/) = (/)
11	6, 9, 10	3eqtr3i 2652	1 |- ( A o. ( B \ `' `' B ) ) = (/)

Syntax hints:   = wceq 1483   \ cdif 3571   (/)c0 3915   `'ccnv 5113   o. ccom 5118   Rel wrel 5119

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  df-co 5123

This theorem is referenced by:  cnvtrcl0  37933