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Theorem nonrel 38207
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 5621 . . 3 𝐴 = (𝐴 ∩ (V × V))
21difeq2i 3758 . 2 (𝐴𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V)))
3 difin 3894 . 2 (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V))
42, 3eqtri 2673 1 (𝐴𝐴) = (𝐴 ∖ (V × V))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  Vcvv 3231  cdif 3604  cin 3606   × cxp 5141  ccnv 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151
This theorem is referenced by:  elnonrel  38208
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