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Theorem sels 5303
 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 4562 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 snex 5301 . . 3 {𝐴} ∈ V
3 eleq2 2878 . . 3 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
42, 3spcev 3556 . 2 (𝐴 ∈ {𝐴} → ∃𝑥 𝐴𝑥)
51, 4syl 17 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1781   ∈ wcel 2111  {csn 4528 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-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-dif 3886  df-un 3888  df-nul 4247  df-sn 4529  df-pr 4531 This theorem is referenced by:  sat1el2xp  32805
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