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Theorem sels 5393
Description: If a class is a set, then it is a member of a set. (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5395. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5254, ax-nul 5261, ax-pow 5318. (Revised by BTernaryTau, 15-Jan-2025.)
Assertion
Ref Expression
sels (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sels
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2825 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21exbidv 1924 . 2 (𝑦 = 𝐴 → (∃𝑥 𝑦𝑥 ↔ ∃𝑥 𝐴𝑥))
3 el 5392 . 2 𝑥 𝑦𝑥
42, 3vtoclg 3523 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wex 1781  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815
This theorem is referenced by:  sat1el2xp  33785
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