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Theorem nonrel 40075
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 6021 . . 3 𝐴 = (𝐴 ∩ (V × V))
21difeq2i 4071 . 2 (𝐴𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V)))
3 difin 4212 . 2 (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V))
42, 3eqtri 2843 1 (𝐴𝐴) = (𝐴 ∖ (V × V))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3470  cdif 3906  cin 3908   × cxp 5525  ccnv 5526
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5039  df-opab 5101  df-xp 5533  df-rel 5534  df-cnv 5535
This theorem is referenced by:  elnonrel  40076
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