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Theorem epnsymrel 38563
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 9649 . . . 4 E ≠ E
21neii 2942 . . 3 ¬ E = E
32intnanr 487 . 2 ¬ ( E = E ∧ Rel E )
4 dfsymrel4 38552 . 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 1540   E cep 5583  ccnv 5684  Rel wrel 5690   SymRel wsymrel 38194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-fr 5637  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-symrel 38545
This theorem is referenced by: (None)
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