Theorem ssext

Description: An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004)

	Ref	Expression
Assertion	ssext	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ssextss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) )
2		ssextss	⊢ ( 𝐵 ⊆ 𝐴 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴 ) )
3	1 2	anbi12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) ↔ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ∧ ∀ 𝑥 ( 𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴 ) ) )
4		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
5		albiim	⊢ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵 ) ↔ ( ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ∧ ∀ 𝑥 ( 𝑥 ⊆ 𝐵 → 𝑥 ⊆ 𝐴 ) ) )
6	3 4 5	3bitr4i	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵 ) )