Theorem nonrel 43616

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 6139	. . 3 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2	1	difeq2i 4073	. 2 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V)))
3		difin 4222	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V))
4	2, 3	eqtri 2754	1 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))

Syntax hints:   = wceq 1541  Vcvv 3436   ∖ cdif 3899   ∩ cin 3901   × cxp 5614  ^◡ccnv 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624

This theorem is referenced by:  elnonrel  43617