Theorem intpr

Description: The intersection of a pair is the intersection of its members. Theorem 71 of Suppes p. 42. (Contributed by NM, 14-Oct-1999) Prove from intprg . (Revised by BJ, 1-Sep-2024)

	Ref	Expression
Hypotheses	intpr.1	⊢ 𝐴 ∈ V
	intpr.2	⊢ 𝐵 ∈ V
Assertion	intpr	⊢ ∩ { 𝐴 , 𝐵 } = ( 𝐴 ∩ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		intpr.1	⊢ 𝐴 ∈ V
2		intpr.2	⊢ 𝐵 ∈ V
3		intprg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ∩ { 𝐴 , 𝐵 } = ( 𝐴 ∩ 𝐵 ) )
4	1 2 3	mp2an	⊢ ∩ { 𝐴 , 𝐵 } = ( 𝐴 ∩ 𝐵 )