Theorem nonrel 44678

Description: A non-relation is equal to the base class with all ordered pairs removed. (Contributed by RP, 25-Oct-2020.)

Assertion

Ref	Expression
nonrel	⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))

Proof of Theorem nonrel

Step	Hyp	Ref	Expression
1		cnvcnv 6183	. . 3 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2	1	difeq2i 4060	. 2 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V)))
3		difin 4217	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V))
4	2, 3	eqtri 2783	1 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3450   ∖ cdif 4045   ∩ cin 4089   × cxp 5648  ^◡ccnv 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  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 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658

This theorem is used by:  elnonrel  44679