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Theorem bj-sels 32143
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 4153 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 sbcel2 3941 . . . 4 ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥)
3 snex 4835 . . . . . 6 {𝐴} ∈ V
4 csbvarg 3955 . . . . . 6 ({𝐴} ∈ V → {𝐴} / 𝑥𝑥 = {𝐴})
53, 4ax-mp 5 . . . . 5 {𝐴} / 𝑥𝑥 = {𝐴}
65eleq2i 2680 . . . 4 (𝐴{𝐴} / 𝑥𝑥𝐴 ∈ {𝐴})
72, 6bitri 263 . . 3 ([{𝐴} / 𝑥]𝐴𝑥𝐴 ∈ {𝐴})
81, 7sylibr 223 . 2 (𝐴𝑉[{𝐴} / 𝑥]𝐴𝑥)
98spesbcd 3488 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wex 1695  wcel 1977  Vcvv 3173  [wsbc 3402  csb 3499  {csn 4125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128
This theorem is referenced by: (None)
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