Theorem cnvnonrel 44266

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 6144	. 2 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = (◡𝐴 ∖ ◡◡◡𝐴)
2		relcnv 6110	. . 3 ⊢ Rel ◡𝐴
3		relnonrel 44265	. . 3 ⊢ (Rel ◡𝐴 ↔ (◡𝐴 ∖ ◡◡◡𝐴) = ∅)
4	2, 3	mpbi 233	. 2 ⊢ (◡𝐴 ∖ ◡◡◡𝐴) = ∅
5	1, 4	eqtri 2793	1 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1568   ∖ cdif 3910  ∅c0 4294  ^◡ccnv 5664  Rel wrel 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-res 5677

This theorem is referenced by:  brnonrel  44267  dmnonrel  44268  resnonrel  44270  cononrel1  44272  cononrel2  44273  clcnvlem  44301  cnvrcl0  44303