Theorem dfss5

Description: Alternate definition of subclass relationship: a class A is a subclass of another class B iff each element of A is equal to an element of B . (Contributed by AV, 13-Nov-2020)

	Ref	Expression
Assertion	dfss5	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		dfss3	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )
2		clel5	⊢ ( 𝑥 ∈ 𝐵 ↔ ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑦 )
3	2	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑦 )
4	1 3	bitri	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑦 )