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Theorem epnsymrel 38544
Description: The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
Assertion
Ref Expression
epnsymrel ¬ SymRel E

Proof of Theorem epnsymrel
StepHypRef Expression
1 epnsym 9647 . . . 4 E ≠ E
21neii 2940 . . 3 ¬ E = E
32intnanr 487 . 2 ¬ ( E = E ∧ Rel E )
4 dfsymrel4 38533 . 2 ( SymRel E ↔ ( E = E ∧ Rel E ))
53, 4mtbir 323 1 ¬ SymRel E
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537   E cep 5588  ccnv 5688  Rel wrel 5694   SymRel wsymrel 38174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-fr 5641  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-symrel 38526
This theorem is referenced by: (None)
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