Theorem 0sn0ep 5556

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 4624	. 2 ⊢ ∅ ∈ {∅}
3		snex 5401	. . 3 ⊢ {∅} ∈ V
4	3	epeli 5554	. 2 ⊢ (∅ E {∅} ↔ ∅ ∈ {∅})
5	2, 4	mpbir 234	1 ⊢ ∅ E {∅}

Syntax hints:   ∈ wcel 2145  ∅c0 4288  {csn 4585   class class class wbr 5105   E cep 5551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-eprel 5552

This theorem is referenced by:  epn0  5557