Theorem epn0 5560

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 5559	. 2 ⊢ ∅ E {∅}
2		brne0 5158	. 2 ⊢ (∅ E {∅} → E ≠ ∅)
3	1, 2	ax-mp 5	1 ⊢ E ≠ ∅

This proof depends on syntax axioms:   ≠ wne 2959  ∅c0 4281  {csn 4588   class class class wbr 5106   E cep 5554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-eprel 5555

This theorem is used by:  epnsym  9623