Theorem cnvnonrel 44778

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 6145	. 2 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = (◡𝐴 ∖ ◡◡◡𝐴)
2		relcnv 6111	. . 3 ⊢ Rel ◡𝐴
3		relnonrel 44777	. . 3 ⊢ (Rel ◡𝐴 ↔ (◡𝐴 ∖ ◡◡◡𝐴) = ∅)
4	2, 3	mpbi 233	. 2 ⊢ (◡𝐴 ∖ ◡◡◡𝐴) = ∅
5	1, 4	eqtri 2786	1 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 4048  ∅c0 4281  ^◡ccnv 5654  Rel wrel 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-res 5667

This theorem is used by:  brnonrel  44779  dmnonrel  44780  resnonrel  44782  cononrel1  44784  cononrel2  44785  clcnvlem  44813  cnvrcl0  44815