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Theorem epnsymrel 37053
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 9552 . . . 4 E ≠ E
21neii 2946 . . 3 ¬ E = E
32intnanr 489 . 2 ¬ ( E = E ∧ Rel E )
4 dfsymrel4 37042 . 2 ( SymRel E ↔ ( E = E ∧ Rel E ))
53, 4mtbir 323 1 ¬ SymRel E
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1542   E cep 5541  ccnv 5637  Rel wrel 5643   SymRel wsymrel 36675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-reg 9535
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-eprel 5542  df-fr 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-symrel 37035
This theorem is referenced by: (None)
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