Theorem 0sn0ep 5499

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 5231	. . 3 ⊢ ∅ ∈ V
2	1	snid 4597	. 2 ⊢ ∅ ∈ {∅}
3		snex 5354	. . 3 ⊢ {∅} ∈ V
4	3	epeli 5497	. 2 ⊢ (∅ E {∅} ↔ ∅ ∈ {∅})
5	2, 4	mpbir 230	1 ⊢ ∅ E {∅}

Syntax hints:   ∈ wcel 2106  ∅c0 4256  {csn 4561   class class class wbr 5074   E cep 5494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-eprel 5495

This theorem is referenced by:  epn0  5500