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Theorem elex2 3417
Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
Assertion
Ref Expression
elex2 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elex2
StepHypRef Expression
1 eleq1a 2853 . . 3 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
21alrimiv 1970 . 2 (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝑥𝐵))
3 elisset 3416 . 2 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
4 exim 1877 . 2 (∀𝑥(𝑥 = 𝐴𝑥𝐵) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥 𝑥𝐵))
52, 3, 4sylc 65 1 (𝐴𝐵 → ∃𝑥 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1599   = wceq 1601  wex 1823  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1605  df-ex 1824  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-v 3399
This theorem is referenced by:  negn0  10804  nocvxmin  32497  itg2addnclem2  34081  risci  34404  dvh1dimat  37589
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