Theorem unipwr

Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw . The proof of this theorem was automatically generated from unipwrVD using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	unipwr	⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2	1	snid	⊢ 𝑥 ∈ { 𝑥 }
3		snelpwi	⊢ ( 𝑥 ∈ 𝐴 → { 𝑥 } ∈ 𝒫 𝐴 )
4		elunii	⊢ ( ( 𝑥 ∈ { 𝑥 } ∧ { 𝑥 } ∈ 𝒫 𝐴 ) → 𝑥 ∈ ∪ 𝒫 𝐴 )
5	2 3 4	sylancr	⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴 )
6	5	ssriv	⊢ 𝐴 ⊆ ∪ 𝒫 𝐴