Theorem elex2 2841

Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) Avoid ax-9 2157, 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 2840	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
2		exsimpr 1902	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐵)
3	1, 2	sylbi 220	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2839

This theorem is used by:  noel  4286  negn0  11784  axprALT2  35857  srepdflem2  36127  itg2addnclem2  38858  risci  39187  dvh1dimat  42764