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Theorem epnsymrel 38090
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 9632 . . . 4 E ≠ E
21neii 2932 . . 3 ¬ E = E
32intnanr 486 . 2 ¬ ( E = E ∧ Rel E )
4 dfsymrel4 38079 . 2 ( SymRel E ↔ ( E = E ∧ Rel E ))
53, 4mtbir 322 1 ¬ SymRel E
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 394   = wceq 1533   E cep 5575  ccnv 5671  Rel wrel 5677   SymRel wsymrel 37717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-reg 9615
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-eprel 5576  df-fr 5627  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-symrel 38072
This theorem is referenced by: (None)
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