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Theorem epn0 5167
 Description: The epsilon relation is not empty. (Contributed by AV, 19-Jun-2022.)
Assertion
Ref Expression
epn0 E ≠ ∅

Proof of Theorem epn0
StepHypRef Expression
1 0sn0ep 5166 . 2 ∅ E {∅}
2 brne0 4834 . 2 (∅ E {∅} → E ≠ ∅)
31, 2ax-mp 5 1 E ≠ ∅
 Colors of variables: wff setvar class Syntax hints:   ≠ wne 2942  ∅c0 4061  {csn 4314   class class class wbr 4784   E cep 5161 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-eprel 5162 This theorem is referenced by:  epnsym  8667
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