Theorem epn0 5537

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 5536	. 2 ⊢ ∅ E {∅}
2		brne0 5150	. 2 ⊢ (∅ E {∅} → E ≠ ∅)
3	1, 2	ax-mp 5	1 ⊢ E ≠ ∅

Syntax hints:   ≠ wne 2933  ∅c0 4287  {csn 4582   class class class wbr 5100   E cep 5531

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-eprel 5532

This theorem is referenced by:  epnsym  9530