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Theorem bj-sels 33256
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 4351 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 sbcel2 4132 . . . 4 ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥)
3 snex 5057 . . . . . 6 {𝐴} ∈ V
4 csbvarg 4146 . . . . . 6 ({𝐴} ∈ V → {𝐴} / 𝑥𝑥 = {𝐴})
53, 4ax-mp 5 . . . . 5 {𝐴} / 𝑥𝑥 = {𝐴}
65eleq2i 2831 . . . 4 (𝐴{𝐴} / 𝑥𝑥𝐴 ∈ {𝐴})
72, 6bitri 264 . . 3 ([{𝐴} / 𝑥]𝐴𝑥𝐴 ∈ {𝐴})
81, 7sylibr 224 . 2 (𝐴𝑉[{𝐴} / 𝑥]𝐴𝑥)
98spesbcd 3663 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  [wsbc 3576  csb 3674  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324
This theorem is referenced by: (None)
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