Theorem sels 5326

Description: If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sels

Step	Hyp	Ref	Expression
1		snidg 4575	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		snex 5324	. . 3 ⊢ {𝐴} ∈ V
3		eleq2 2826	. . 3 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
4	2, 3	spcev 3521	. 2 ⊢ (𝐴 ∈ {𝐴} → ∃𝑥 𝐴 ∈ 𝑥)
5	1, 4	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

Syntax hints:   → wi 4  ∃wex 1787   ∈ wcel 2110  {csn 4541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-sn 4542  df-pr 4544

This theorem is referenced by:  sat1el2xp  33054