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Theorem nonrel 43014
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 (𝐴𝐴) = (𝐴 ∖ (V × V))

Proof of Theorem nonrel
StepHypRef Expression
1 cnvcnv 6196 . . 3 𝐴 = (𝐴 ∩ (V × V))
21difeq2i 4117 . 2 (𝐴𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V)))
3 difin 4262 . 2 (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V))
42, 3eqtri 2756 1 (𝐴𝐴) = (𝐴 ∖ (V × V))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  Vcvv 3471  cdif 3944  cin 3946   × cxp 5676  ccnv 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686
This theorem is referenced by:  elnonrel  43015
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