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Theorem sels 5326
Description: If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)
Assertion
Ref Expression
sels (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sels
StepHypRef Expression
1 snidg 4575 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 snex 5324 . . 3 {𝐴} ∈ V
3 eleq2 2826 . . 3 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
42, 3spcev 3521 . 2 (𝐴 ∈ {𝐴} → ∃𝑥 𝐴𝑥)
51, 4syl 17 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1787  wcel 2110  {csn 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-sn 4542  df-pr 4544
This theorem is referenced by:  sat1el2xp  33054
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