MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elex2 Structured version   Visualization version   GIF version

Theorem elex2 2818
Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) Avoid ax-9 2118, ax-ext 2708, df-clab 2715. (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 2817 . 2 (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
2 exsimpr 1869 . 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 1540  wex 1779  wcel 2108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-clel 2816
This theorem is referenced by:  negn0  11692  nocvxmin  27823  itg2addnclem2  37679  risci  37994  dvh1dimat  41443
  Copyright terms: Public domain W3C validator