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Theorem bj-sels 33252
 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 4347 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 sbcel2 4128 . . . 4 ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥)
3 snex 5053 . . . . . 6 {𝐴} ∈ V
4 csbvarg 4142 . . . . . 6 ({𝐴} ∈ V → {𝐴} / 𝑥𝑥 = {𝐴})
53, 4ax-mp 5 . . . . 5 {𝐴} / 𝑥𝑥 = {𝐴}
65eleq2i 2827 . . . 4 (𝐴{𝐴} / 𝑥𝑥𝐴 ∈ {𝐴})
72, 6bitri 264 . . 3 ([{𝐴} / 𝑥]𝐴𝑥𝐴 ∈ {𝐴})
81, 7sylibr 224 . 2 (𝐴𝑉[{𝐴} / 𝑥]𝐴𝑥)
98spesbcd 3659 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1628  ∃wex 1849   ∈ wcel 2135  Vcvv 3336  [wsbc 3572  ⦋csb 3670  {csn 4317 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-fal 1634  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-nul 4055  df-sn 4318  df-pr 4320 This theorem is referenced by: (None)
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