Theorem nonrel 43573

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 6213	. . 3 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2	1	difeq2i 4132	. 2 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V)))
3		difin 4277	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V))
4	2, 3	eqtri 2762	1 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))

Syntax hints:   = wceq 1536  Vcvv 3477   ∖ cdif 3959   ∩ cin 3961   × cxp 5686  ^◡ccnv 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696

This theorem is referenced by:  elnonrel  43574