Theorem cnvnonrel 44532

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 6128	. 2 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = (◡𝐴 ∖ ◡◡◡𝐴)
2		relcnv 6094	. . 3 ⊢ Rel ◡𝐴
3		relnonrel 44531	. . 3 ⊢ (Rel ◡𝐴 ↔ (◡𝐴 ∖ ◡◡◡𝐴) = ∅)
4	2, 3	mpbi 233	. 2 ⊢ (◡𝐴 ∖ ◡◡◡𝐴) = ∅
5	1, 4	eqtri 2783	1 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3895  ∅c0 4278  ^◡ccnv 5646  Rel wrel 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-res 5659

This theorem is used by:  brnonrel  44533  dmnonrel  44534  resnonrel  44536  cononrel1  44538  cononrel2  44539  clcnvlem  44567  cnvrcl0  44569