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Theorem elex2 3247
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 2725 . . 3 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
21alrimiv 1895 . 2 (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝑥𝐵))
3 elisset 3246 . 2 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
4 exim 1801 . 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 1521   = wceq 1523  wex 1744  wcel 2030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1526  df-ex 1745  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233
This theorem is referenced by:  negn0  10497  nocvxmin  32019  itg2addnclem2  33592  risci  33916  dvh1dimat  37047
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