Theorem 0sn0ep 5542

Description: An example for the membership relation. (Contributed by AV, 19-Jun-2022.)

Assertion

Ref	Expression
0sn0ep	⊢ ∅ E {∅}

Proof of Theorem 0sn0ep

Step	Hyp	Ref	Expression
1		0ex 5262	. . 3 ⊢ ∅ ∈ V
2	1	snid 4626	. 2 ⊢ ∅ ∈ {∅}
3		snex 5391	. . 3 ⊢ {∅} ∈ V
4	3	epeli 5540	. 2 ⊢ (∅ E {∅} ↔ ∅ ∈ {∅})
5	2, 4	mpbir 231	1 ⊢ ∅ E {∅}

Syntax hints:   ∈ wcel 2109  ∅c0 4296  {csn 4589   class class class wbr 5107   E cep 5537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-eprel 5538

This theorem is referenced by:  epn0  5543