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Theorem bj-sels 33530
 Description: If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)
Assertion
Ref Expression
bj-sels (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-sels
StepHypRef Expression
1 snidg 4428 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 snex 5142 . . 3 {𝐴} ∈ V
3 eleq2 2848 . . 3 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
42, 3spcev 3502 . 2 (𝐴 ∈ {𝐴} → ∃𝑥 𝐴𝑥)
51, 4syl 17 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1823   ∈ wcel 2107  {csn 4398 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-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-dif 3795  df-un 3797  df-nul 4142  df-sn 4399  df-pr 4401 This theorem is referenced by: (None)
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