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 5155	. 2 ⊢ (∅ E {∅} → E ≠ ∅)
3	1, 2	ax-mp 5	1 ⊢ E ≠ ∅

This proof depends on syntax axioms:   ≠ wne 2955  ∅c0 4279  {csn 4584   class class class wbr 5103   E cep 5554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-eprel 5555

This theorem is used by:  epnsym  9591