Theorem ifpr

Description: Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007)

	Ref	Expression
Assertion	ifpr	$$⊢\left(A\in C\wedge B\in D\right)\to if\left(\phi ,A,B\right)\in \left\{A,B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in C\to A\in \mathrm{V}$$
2		elex	$$⊢B\in D\to B\in \mathrm{V}$$
3		ifcl	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to if\left(\phi ,A,B\right)\in \mathrm{V}$$
4		ifeqor	$$⊢\left(if\left(\phi ,A,B\right)=A\vee if\left(\phi ,A,B\right)=B\right)$$
5		elprg	$$⊢if\left(\phi ,A,B\right)\in \mathrm{V}\to \left(if\left(\phi ,A,B\right)\in \left\{A,B\right\}\leftrightarrow \left(if\left(\phi ,A,B\right)=A\vee if\left(\phi ,A,B\right)=B\right)\right)$$
6	4 5	mpbiri	$$⊢if\left(\phi ,A,B\right)\in \mathrm{V}\to if\left(\phi ,A,B\right)\in \left\{A,B\right\}$$
7	3 6	syl	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to if\left(\phi ,A,B\right)\in \left\{A,B\right\}$$
8	1 2 7	syl2an	$$⊢\left(A\in C\wedge B\in D\right)\to if\left(\phi ,A,B\right)\in \left\{A,B\right\}$$