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 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ancom	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) )
2		elin	⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
3		elin	⊢ ( 𝑥 ∈ ( 𝐵 ∩ 𝐴 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) )
4	1 2 3	3bitr4i	⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐵 ) ↔ 𝑥 ∈ ( 𝐵 ∩ 𝐴 ) )
5	4	eqriv	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )