Theorem nonrel 44195

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 6189	. . 3 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2	1	difeq2i 4086	. 2 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V)))
3		difin 4233	. 2 ⊢ (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V))
4	2, 3	eqtri 2792	1 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))

Syntax hints:   = wceq 1567  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912   × cxp 5657  ^◡ccnv 5658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667

This theorem is referenced by:  elnonrel  44196