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Theorem bj-sels 10034
 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 3400 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 bj-snexg 10032 . . . . 5 (𝐴𝑉 → {𝐴} ∈ V)
3 sbcel2g 2871 . . . . 5 ({𝐴} ∈ V → ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥))
42, 3syl 14 . . . 4 (𝐴𝑉 → ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥))
5 csbvarg 2877 . . . . . 6 ({𝐴} ∈ V → {𝐴} / 𝑥𝑥 = {𝐴})
62, 5syl 14 . . . . 5 (𝐴𝑉{𝐴} / 𝑥𝑥 = {𝐴})
76eleq2d 2107 . . . 4 (𝐴𝑉 → (𝐴{𝐴} / 𝑥𝑥𝐴 ∈ {𝐴}))
84, 7bitrd 177 . . 3 (𝐴𝑉 → ([{𝐴} / 𝑥]𝐴𝑥𝐴 ∈ {𝐴}))
91, 8mpbird 156 . 2 (𝐴𝑉[{𝐴} / 𝑥]𝐴𝑥)
109spesbcd 2844 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  [wsbc 2764  ⦋csb 2852  {csn 3375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-pr 3944  ax-bdor 9936  ax-bdeq 9940  ax-bdsep 10004 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-sn 3381  df-pr 3382 This theorem is referenced by: (None)
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