Theorem elssuni

Description: An element of a class is a subclass of its union. Theorem 8.6 of Quine p. 54. Also the basis for Proposition 7.20 of TakeutiZaring p. 40. (Contributed by NM, 6-Jun-1994)

	Ref	Expression
Assertion	elssuni	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ssid	⊢ 𝐴 ⊆ 𝐴
2		ssuni	⊢ ( ( 𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵 ) → 𝐴 ⊆ ∪ 𝐵 )
3	1 2	mpan	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵 )