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Theorem epn0 5543
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 5542 . 2 ∅ E {∅}
2 brne0 5156 . 2 (∅ E {∅} → E ≠ ∅)
31, 2ax-mp 5 1 E ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 2944  c0 4283  {csn 4587   class class class wbr 5106   E cep 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-eprel 5538
This theorem is referenced by:  epnsym  9546
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