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Theorem epnsymrel 35907
 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 9069 . . . 4 E ≠ E
21neii 3016 . . 3 ¬ E = E
32intnanr 491 . 2 ¬ ( E = E ∧ Rel E )
4 dfsymrel4 35896 . 2 ( SymRel E ↔ ( E = E ∧ Rel E ))
53, 4mtbir 326 1 ¬ SymRel E
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538   E cep 5451  ◡ccnv 5541  Rel wrel 5547   SymRel wsymrel 35574 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-reg 9053 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-eprel 5452  df-fr 5501  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-symrel 35889 This theorem is referenced by: (None)
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