Theorem sels 5419

Description: If a class is a set, then it is a member of a set. (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5421. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5255, ax-nul 5267, ax-pow 5334. (Revised by BTernaryTau, 15-Jan-2025.)

Assertion

Ref	Expression
sels	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sels

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2850	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2	1	exbidv 1954	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝐴 ∈ 𝑥))
3		el 5417	. 2 ⊢ ∃𝑥 𝑦 ∈ 𝑥
4	2, 3	vtoclg 3520	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2145

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837

This theorem is used by:  sat1el2xp  35966