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Theorem elex2 3491
 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 2909 . . 3 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
21alrimiv 1928 . 2 (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝑥𝐵))
3 elisset 3480 . 2 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
4 exim 1835 . 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 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2114 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2815  df-clel 2894 This theorem is referenced by:  negn0  11058  nocvxmin  33322  itg2addnclem2  35067  risci  35383  dvh1dimat  38695
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