Theorem 0sn0ep 5167

Description: An example of an epsilon relation. (Contributed by AV, 19-Jun-2022.)

Assertion

Ref	Expression
0sn0ep	⊢ ∅ E {∅}

Proof of Theorem 0sn0ep

Step	Hyp	Ref	Expression
1		0ex 4925	. . 3 ⊢ ∅ ∈ V
2	1	snid 4348	. 2 ⊢ ∅ ∈ {∅}
3		snex 5037	. . 3 ⊢ {∅} ∈ V
4	3	epelc 5165	. 2 ⊢ (∅ E {∅} ↔ ∅ ∈ {∅})
5	2, 4	mpbir 221	1 ⊢ ∅ E {∅}

Syntax hints:   ∈ wcel 2145  ∅c0 4063  {csn 4317   class class class wbr 4787   E cep 5162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-eprel 5163

This theorem is referenced by:  epn0  5168