Theorem 0sn0ep 5434

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 5175	. . 3 ⊢ ∅ ∈ V
2	1	snid 4561	. 2 ⊢ ∅ ∈ {∅}
3		snex 5297	. . 3 ⊢ {∅} ∈ V
4	3	epeli 5432	. 2 ⊢ (∅ E {∅} ↔ ∅ ∈ {∅})
5	2, 4	mpbir 234	1 ⊢ ∅ E {∅}

Syntax hints:   ∈ wcel 2111  ∅c0 4243  {csn 4525   class class class wbr 5030   E cep 5429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-eprel 5430

This theorem is referenced by:  epn0  5435