Theorem dfss2OLD

Description: Obsolete version of dfss2 as of 16-May-2024. (Contributed by NM, 8-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	dfss2OLD	$$⊢A\subseteq B\leftrightarrow \forall x\left(x\in A\to x\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfss	$$⊢A\subseteq B\leftrightarrow A=A\cap B$$
2		df-in	$$⊢A\cap B=\left\{x|\left(x\in A\wedge x\in B\right)\right\}$$
3	2	eqeq2i	$$⊢A=A\cap B\leftrightarrow A=\left\{x|\left(x\in A\wedge x\in B\right)\right\}$$
4		abeq2	$$⊢A=\left\{x|\left(x\in A\wedge x\in B\right)\right\}\leftrightarrow \forall x\left(x\in A\leftrightarrow \left(x\in A\wedge x\in B\right)\right)$$
5	1 3 4	3bitri	$$⊢A\subseteq B\leftrightarrow \forall x\left(x\in A\leftrightarrow \left(x\in A\wedge x\in B\right)\right)$$
6		pm4.71	$$⊢\left(x\in A\to x\in B\right)\leftrightarrow \left(x\in A\leftrightarrow \left(x\in A\wedge x\in B\right)\right)$$
7	6	albii	$$⊢\forall x\left(x\in A\to x\in B\right)\leftrightarrow \forall x\left(x\in A\leftrightarrow \left(x\in A\wedge x\in B\right)\right)$$
8	5 7	bitr4i	$$⊢A\subseteq B\leftrightarrow \forall x\left(x\in A\to x\in B\right)$$