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Theorem epnsym 8801
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 8800 . 2 ( E ∩ E ) = ∅
2 disjeq0 4248 . 2 (( E ∩ E ) = ∅ → ( E = E ↔ ( E = ∅ ∧ E = ∅)))
3 epn0 5271 . . . . . 6 E ≠ ∅
4 eqneqall 2980 . . . . . 6 ( E = ∅ → ( E ≠ ∅ → E ≠ E ))
53, 4mpi 20 . . . . 5 ( E = ∅ → E ≠ E )
65adantl 475 . . . 4 (( E = ∅ ∧ E = ∅) → E ≠ E )
76a1i 11 . . 3 ( E = E → (( E = ∅ ∧ E = ∅) → E ≠ E ))
8 neqne 2977 . . . 4 E = E → E ≠ E )
98a1d 25 . . 3 E = E → (¬ ( E = ∅ ∧ E = ∅) → E ≠ E ))
107, 9bija 372 . 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 198  wa 386   = wceq 1601  wne 2969  cin 3791  c0 4141   E cep 5265  ccnv 5354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-reg 8786
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-eprel 5266  df-fr 5314  df-xp 5361  df-rel 5362  df-cnv 5363
This theorem is referenced by:  epnsymrel  34936
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