Theorem cnvnonrel 41196

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 6047	. 2 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = (◡𝐴 ∖ ◡◡◡𝐴)
2		relcnv 6012	. . 3 ⊢ Rel ◡𝐴
3		relnonrel 41195	. . 3 ⊢ (Rel ◡𝐴 ↔ (◡𝐴 ∖ ◡◡◡𝐴) = ∅)
4	2, 3	mpbi 229	. 2 ⊢ (◡𝐴 ∖ ◡◡◡𝐴) = ∅
5	1, 4	eqtri 2766	1 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1539   ∖ cdif 3884  ∅c0 4256  ^◡ccnv 5588  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597

This theorem is referenced by:  brnonrel  41197  dmnonrel  41198  resnonrel  41200  cononrel1  41202  cononrel2  41203  clcnvlem  41231  cnvrcl0  41233