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Theorem 0sn0ep 5270
 Description: An example for the membership relation. (Contributed by AV, 19-Jun-2022.)
Assertion
Ref Expression
0sn0ep ∅ E {∅}

Proof of Theorem 0sn0ep
StepHypRef Expression
1 0ex 5026 . . 3 ∅ ∈ V
21snid 4430 . 2 ∅ ∈ {∅}
3 snex 5140 . . 3 {∅} ∈ V
43epeli 5268 . 2 (∅ E {∅} ↔ ∅ ∈ {∅})
52, 4mpbir 223 1 ∅ E {∅}
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2107  ∅c0 4141  {csn 4398   class class class wbr 4886   E cep 5265 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-eprel 5266 This theorem is referenced by:  epn0  5271
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