Theorem cnvnonrel 37894

Description: The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.)

Assertion

Ref	Expression
cnvnonrel	|- `' ( A \ `' `' A ) = (/)

Proof of Theorem cnvnonrel

Step	Hyp	Ref	Expression
1		cnvdif 5539	. 2 |- `' ( A \ `' `' A ) = ( `' A \ `' `' `' A )
2		relcnv 5503	. . 3 |- Rel `' A
3		relnonrel 37893	. . 3 |- ( Rel `' A <-> ( `' A \ `' `' `' A ) = (/) )
4	2, 3	mpbi 220	. 2 |- ( `' A \ `' `' `' A ) = (/)
5	1, 4	eqtri 2644	1 |- `' ( A \ `' `' A ) = (/)

Syntax hints:   = wceq 1483   \ cdif 3571   (/)c0 3915   `'ccnv 5113   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

This theorem is referenced by:  brnonrel  37895  dmnonrel  37896  resnonrel  37898  cononrel1  37900  cononrel2  37901  clcnvlem  37930  cnvrcl0  37932