Theorem nonrel 43580

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 6168	. . 3 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2	1	difeq2i 4089	. 2 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V)))
3		difin 4238	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V))
4	2, 3	eqtri 2753	1 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))

Syntax hints:   = wceq 1540  Vcvv 3450   ∖ cdif 3914   ∩ cin 3916   × cxp 5639  ^◡ccnv 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649

This theorem is referenced by:  elnonrel  43581