Theorem sels 5407

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 5409. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5246, ax-nul 5256, ax-pow 5322. (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 1941	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝐴 ∈ 𝑥))
3		el 5405	. 2 ⊢ ∃𝑥 𝑦 ∈ 𝑥
4	2, 3	vtoclg 3522	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4   = wceq 1560  ∃wex 1799   ∈ wcel 2142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837

This theorem is referenced by:  sat1el2xp  35729