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Theorem epnsym 9069
 Description: The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
Assertion
Ref Expression
epnsym E ≠ E

Proof of Theorem epnsym
StepHypRef Expression
1 cnvepnep 9068 . 2 ( E ∩ E ) = ∅
2 disjeq0 4388 . 2 (( E ∩ E ) = ∅ → ( E = E ↔ ( E = ∅ ∧ E = ∅)))
3 epn0 5458 . . . . . 6 E ≠ ∅
4 eqneqall 3025 . . . . . 6 ( E = ∅ → ( E ≠ ∅ → E ≠ E ))
53, 4mpi 20 . . . . 5 ( E = ∅ → E ≠ E )
65adantl 485 . . . 4 (( E = ∅ ∧ E = ∅) → E ≠ E )
76a1i 11 . . 3 ( E = E → (( E = ∅ ∧ E = ∅) → E ≠ E ))
8 neqne 3022 . . . 4 E = E → E ≠ E )
98a1d 25 . . 3 E = E → (¬ ( E = ∅ ∧ E = ∅) → E ≠ E ))
107, 9bija 385 . 2 (( E = E ↔ ( E = ∅ ∧ E = ∅)) → E ≠ E )
111, 2, 10mp2b 10 1 E ≠ E
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ≠ wne 3014   ∩ cin 3918  ∅c0 4276   E cep 5451  ◡ccnv 5541 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 This theorem is referenced by:  epnsymrel  35903
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