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	$$⊢A\in \mathrm{V}$$
	intpr.2	$$⊢B\in \mathrm{V}$$
Assertion	intpr	$$⊢\bigcap \left\{A,B\right\}=A\cap B$$

Proof

Step	Hyp	Ref	Expression
1		intpr.1	$$⊢A\in \mathrm{V}$$
2		intpr.2	$$⊢B\in \mathrm{V}$$
3		intprg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \bigcap \left\{A,B\right\}=A\cap B$$
4	1 2 3	mp2an	$$⊢\bigcap \left\{A,B\right\}=A\cap B$$