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	$$⊢A=B\leftrightarrow \forall x\left(x\subseteq A\leftrightarrow x\subseteq B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssextss	$$⊢A\subseteq B\leftrightarrow \forall x\left(x\subseteq A\to x\subseteq B\right)$$
2		ssextss	$$⊢B\subseteq A\leftrightarrow \forall x\left(x\subseteq B\to x\subseteq A\right)$$
3	1 2	anbi12i	$$⊢\left(A\subseteq B\wedge B\subseteq A\right)\leftrightarrow \left(\forall x\left(x\subseteq A\to x\subseteq B\right)\wedge \forall x\left(x\subseteq B\to x\subseteq A\right)\right)$$
4		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
5		albiim	$$⊢\forall x\left(x\subseteq A\leftrightarrow x\subseteq B\right)\leftrightarrow \left(\forall x\left(x\subseteq A\to x\subseteq B\right)\wedge \forall x\left(x\subseteq B\to x\subseteq A\right)\right)$$
6	3 4 5	3bitr4i	$$⊢A=B\leftrightarrow \forall x\left(x\subseteq A\leftrightarrow x\subseteq B\right)$$