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Theorem bj-sels 10848
Description: If a class is a set, then it is a member of a set. (Copied from set.mm.) (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 3425 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 bj-snexg 10846 . . . . 5 (𝐴𝑉 → {𝐴} ∈ V)
3 sbcel2g 2928 . . . . 5 ({𝐴} ∈ V → ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥))
42, 3syl 14 . . . 4 (𝐴𝑉 → ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥))
5 csbvarg 2934 . . . . . 6 ({𝐴} ∈ V → {𝐴} / 𝑥𝑥 = {𝐴})
62, 5syl 14 . . . . 5 (𝐴𝑉{𝐴} / 𝑥𝑥 = {𝐴})
76eleq2d 2149 . . . 4 (𝐴𝑉 → (𝐴{𝐴} / 𝑥𝑥𝐴 ∈ {𝐴}))
84, 7bitrd 186 . . 3 (𝐴𝑉 → ([{𝐴} / 𝑥]𝐴𝑥𝐴 ∈ {𝐴}))
91, 8mpbird 165 . 2 (𝐴𝑉[{𝐴} / 𝑥]𝐴𝑥)
109spesbcd 2901 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wex 1422  wcel 1434  Vcvv 2602  [wsbc 2816  csb 2909  {csn 3400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-pr 3966  ax-bdor 10750  ax-bdeq 10754  ax-bdsep 10818
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-sn 3406  df-pr 3407
This theorem is referenced by: (None)
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