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Theorem elex2 2816
Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) Avoid ax-9 2116, ax-ext 2706, df-clab 2713. (Revised by Wolf Lammen, 30-Nov-2024.)
Assertion
Ref Expression
elex2 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elex2
StepHypRef Expression
1 dfclel 2815 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
2 exsimpr 1867 . 2 (∃𝑥(𝑥 = 𝐴𝑥𝐵) → ∃𝑥 𝑥𝐵)
31, 2sylbi 217 1 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = 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
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-clel 2814
This theorem is referenced by:  negn0  11690  nocvxmin  27838  itg2addnclem2  37659  risci  37974  dvh1dimat  41424
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