Theorem incomOLD

Description: Obsolete version of incom as of 12-Dec-2023. Commutative law for intersection of classes. Exercise 7 of TakeutiZaring p. 17. (Contributed by NM, 21-Jun-1993) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	incomOLD	$$⊢A\cap B=B\cap A$$

Proof

Step	Hyp	Ref	Expression
1		ancom	$$⊢\left(x\in A\wedge x\in B\right)\leftrightarrow \left(x\in B\wedge x\in A\right)$$
2		elin	$$⊢x\in \left(A\cap B\right)\leftrightarrow \left(x\in A\wedge x\in B\right)$$
3		elin	$$⊢x\in \left(B\cap A\right)\leftrightarrow \left(x\in B\wedge x\in A\right)$$
4	1 2 3	3bitr4i	$$⊢x\in \left(A\cap B\right)\leftrightarrow x\in \left(B\cap A\right)$$
5	4	eqriv	$$⊢A\cap B=B\cap A$$