Theorem epn0 5499

Description: The membership relation is nonempty. (Contributed by AV, 19-Jun-2022.)

Assertion

Ref	Expression
epn0	⊢ E ≠ ∅

Proof of Theorem epn0

Step	Hyp	Ref	Expression
1		0sn0ep 5498	. 2 ⊢ ∅ E {∅}
2		brne0 5128	. 2 ⊢ (∅ E {∅} → E ≠ ∅)
3	1, 2	ax-mp 5	1 ⊢ E ≠ ∅

Syntax hints:   ≠ wne 2944  ∅c0 4261  {csn 4566   class class class wbr 5078   E cep 5493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-eprel 5494

This theorem is referenced by:  epnsym  9328