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Theorem bj-sels 31974
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 4056 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 sbcel2 3844 . . . 4 ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥)
3 snex 4734 . . . . . 6 {𝐴} ∈ V
4 csbvarg 3858 . . . . . 6 ({𝐴} ∈ V → {𝐴} / 𝑥𝑥 = {𝐴})
53, 4ax-mp 5 . . . . 5 {𝐴} / 𝑥𝑥 = {𝐴}
65eleq2i 2584 . . . 4 (𝐴{𝐴} / 𝑥𝑥𝐴 ∈ {𝐴})
72, 6bitri 262 . . 3 ([{𝐴} / 𝑥]𝐴𝑥𝐴 ∈ {𝐴})
81, 7sylibr 222 . 2 (𝐴𝑉[{𝐴} / 𝑥]𝐴𝑥)
98spesbcd 3392 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wex 1694  wcel 1938  Vcvv 3077  [wsbc 3306  csb 3403  {csn 4028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-nul 3778  df-sn 4029  df-pr 4031
This theorem is referenced by: (None)
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