Theorem unipwrVD

Description: Virtual deduction proof of unipwr . (Contributed by Alan Sare, 25-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	unipwrVD	⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2	1	snid	⊢ 𝑥 ∈ { 𝑥 }
3		idn1	⊢ (    𝑥 ∈ 𝐴    ▶    𝑥 ∈ 𝐴    )
4		snelpwi	⊢ ( 𝑥 ∈ 𝐴 → { 𝑥 } ∈ 𝒫 𝐴 )
5	3 4	e1a	⊢ (    𝑥 ∈ 𝐴    ▶    { 𝑥 } ∈ 𝒫 𝐴    )
6		elunii	⊢ ( ( 𝑥 ∈ { 𝑥 } ∧ { 𝑥 } ∈ 𝒫 𝐴 ) → 𝑥 ∈ ∪ 𝒫 𝐴 )
7	2 5 6	e01an	⊢ (    𝑥 ∈ 𝐴    ▶    𝑥 ∈ ∪ 𝒫 𝐴    )
8	7	in1	⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴 )
9	8	ssriv	⊢ 𝐴 ⊆ ∪ 𝒫 𝐴