Theorem elex2 2840

Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) Avoid ax-9 2153, ax-ext 2735, df-clab 2742. (Revised by Wolf Lammen, 30-Nov-2024.)

Assertion

Ref	Expression
elex2	⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elex2

Step	Hyp	Ref	Expression
1		dfclel 2839	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
2		exsimpr 1899	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐵)
3	1, 2	sylbi 220	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-clel 2838

This theorem is referenced by:  negn0  11638  axprALT2  35503  itg2addnclem2  38323  risci  38638  dvh1dimat  42215