Theorem ssun1

Description: Subclass relationship for union of classes. Theorem 25 of Suppes p. 27. (Contributed by NM, 5-Aug-1993)

	Ref	Expression
Assertion	ssun1	⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		orc	⊢ ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) )
2		elun	⊢ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) )
3	1 2	sylibr	⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) )
4	3	ssriv	⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 )