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Theorem bj-sels 13039
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 3524 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 bj-snexg 13037 . . . . 5 (𝐴𝑉 → {𝐴} ∈ V)
3 sbcel2g 2994 . . . . 5 ({𝐴} ∈ V → ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥))
42, 3syl 14 . . . 4 (𝐴𝑉 → ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥))
5 csbvarg 3000 . . . . . 6 ({𝐴} ∈ V → {𝐴} / 𝑥𝑥 = {𝐴})
62, 5syl 14 . . . . 5 (𝐴𝑉{𝐴} / 𝑥𝑥 = {𝐴})
76eleq2d 2187 . . . 4 (𝐴𝑉 → (𝐴{𝐴} / 𝑥𝑥𝐴 ∈ {𝐴}))
84, 7bitrd 187 . . 3 (𝐴𝑉 → ([{𝐴} / 𝑥]𝐴𝑥𝐴 ∈ {𝐴}))
91, 8mpbird 166 . 2 (𝐴𝑉[{𝐴} / 𝑥]𝐴𝑥)
109spesbcd 2967 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660  [wsbc 2882  csb 2975  {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-pr 4101  ax-bdor 12941  ax-bdeq 12945  ax-bdsep 13009
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by: (None)
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