Theorem pwuni

Description: A class is a subclass of the power class of its union. Exercise 6(b) of Enderton p. 38. (Contributed by NM, 14-Oct-1996)

	Ref	Expression
Assertion	pwuni	⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴

Proof

Step	Hyp	Ref	Expression
1		elssuni	⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴 )
2		velpw	⊢ ( 𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴 )
3	1 2	sylibr	⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴 )
4	3	ssriv	⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴