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Theorem epn0 5584
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 5583 . 2 ∅ E {∅}
2 brne0 5197 . 2 (∅ E {∅} → E ≠ ∅)
31, 2ax-mp 5 1 E ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wne 2940  c0 4321  {csn 4627   class class class wbr 5147   E cep 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-eprel 5579
This theorem is referenced by:  epnsym  9600
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