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Theorem nonrel 42885
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 6182 . . 3 𝐴 = (𝐴 ∩ (V × V))
21difeq2i 4112 . 2 (𝐴𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V)))
3 difin 4254 . 2 (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V))
42, 3eqtri 2752 1 (𝐴𝐴) = (𝐴 ∖ (V × V))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3466  cdif 3938  cin 3940   × cxp 5665  ccnv 5666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675
This theorem is referenced by:  elnonrel  42886
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