Theorem cnvnonrel 44342

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 6139	. 2 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = (◡𝐴 ∖ ◡◡◡𝐴)
2		relcnv 6105	. . 3 ⊢ Rel ◡𝐴
3		relnonrel 44341	. . 3 ⊢ (Rel ◡𝐴 ↔ (◡𝐴 ∖ ◡◡◡𝐴) = ∅)
4	2, 3	mpbi 233	. 2 ⊢ (◡𝐴 ∖ ◡◡◡𝐴) = ∅
5	1, 4	eqtri 2785	1 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1569   ∖ cdif 3901  ∅c0 4285  ^◡ccnv 5659  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-res 5672

This theorem is used by:  brnonrel  44343  dmnonrel  44344  resnonrel  44346  cononrel1  44348  cononrel2  44349  clcnvlem  44377  cnvrcl0  44379