Theorem epn0 5604

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 5603	. 2 ⊢ ∅ E {∅}
2		brne0 5216	. 2 ⊢ (∅ E {∅} → E ≠ ∅)
3	1, 2	ax-mp 5	1 ⊢ E ≠ ∅

Syntax hints:   ≠ wne 2946  ∅c0 4352  {csn 4648   class class class wbr 5166   E cep 5598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599

This theorem is referenced by:  epnsym  9678