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Theorem epn0 5582
Description: The membership relation is nonempty. (Contributed by AV, 19-Jun-2022.)
Assertion
Ref Expression
epn0 E ≠ ∅

Proof of Theorem epn0
StepHypRef Expression
1 0sn0ep 5581 . 2 ∅ E {∅}
2 brne0 5193 . 2 (∅ E {∅} → E ≠ ∅)
31, 2ax-mp 5 1 E ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 2936  c0 4319  {csn 4625   class class class wbr 5143   E cep 5576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-eprel 5577
This theorem is referenced by:  epnsym  9627
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