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Theorem sels 5449
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 5451. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5302, ax-nul 5312, ax-pow 5371. (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 2827 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21exbidv 1919 . 2 (𝑦 = 𝐴 → (∃𝑥 𝑦𝑥 ↔ ∃𝑥 𝐴𝑥))
3 el 5448 . 2 𝑥 𝑦𝑥
42, 3vtoclg 3554 1 (𝐴𝑉 → ∃𝑥 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wex 1776  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814
This theorem is referenced by:  sat1el2xp  35364
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