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Theorem epnsym 9601
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 9600 . 2 ( E ∩ E ) = ∅
2 disjeq0 4448 . 2 (( E ∩ E ) = ∅ → ( E = E ↔ ( E = ∅ ∧ E = ∅)))
3 epn0 5576 . . . . . 6 E ≠ ∅
4 eqneqall 2943 . . . . . 6 ( E = ∅ → ( E ≠ ∅ → E ≠ E ))
53, 4mpi 20 . . . . 5 ( E = ∅ → E ≠ E )
65adantl 481 . . . 4 (( E = ∅ ∧ E = ∅) → E ≠ E )
76a1i 11 . . 3 ( E = E → (( E = ∅ ∧ E = ∅) → E ≠ E ))
8 neqne 2940 . . . 4 E = E → E ≠ E )
98a1d 25 . . 3 E = E → (¬ ( E = ∅ ∧ E = ∅) → E ≠ E ))
107, 9bija 380 . 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 205  wa 395   = wceq 1533  wne 2932  cin 3940  c0 4315   E cep 5570  ccnv 5666
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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-reg 9584
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-eprel 5571  df-fr 5622  df-xp 5673  df-rel 5674  df-cnv 5675
This theorem is referenced by:  epnsymrel  37936
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