Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-sels GIF version

Theorem bj-sels 13101
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 3549 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 bj-snexg 13099 . . . . 5 (𝐴𝑉 → {𝐴} ∈ V)
3 sbcel2g 3018 . . . . 5 ({𝐴} ∈ V → ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥))
42, 3syl 14 . . . 4 (𝐴𝑉 → ([{𝐴} / 𝑥]𝐴𝑥𝐴{𝐴} / 𝑥𝑥))
5 csbvarg 3025 . . . . . 6 ({𝐴} ∈ V → {𝐴} / 𝑥𝑥 = {𝐴})
62, 5syl 14 . . . . 5 (𝐴𝑉{𝐴} / 𝑥𝑥 = {𝐴})
76eleq2d 2207 . . . 4 (𝐴𝑉 → (𝐴{𝐴} / 𝑥𝑥𝐴 ∈ {𝐴}))
84, 7bitrd 187 . . 3 (𝐴𝑉 → ([{𝐴} / 𝑥]𝐴𝑥𝐴 ∈ {𝐴}))
91, 8mpbird 166 . 2 (𝐴𝑉[{𝐴} / 𝑥]𝐴𝑥)
109spesbcd 2990 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wex 1468  wcel 1480  Vcvv 2681  [wsbc 2904  csb 2998  {csn 3522
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-pr 4126  ax-bdor 13003  ax-bdeq 13007  ax-bdsep 13071
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-sn 3528  df-pr 3529
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator