Theorem umgrbi

Description: Show that an unordered pair is a valid edge in a multigraph. (Contributed by AV, 9-Mar-2021)

	Ref	Expression
Hypotheses	umgrbi.x	$$⊢X\in V$$
	umgrbi.y	$$⊢Y\in V$$
	umgrbi.n	$$⊢X\ne Y$$
Assertion	umgrbi	$$⊢\left\{X,Y\right\}\in \left\{x\in 𝒫V|\left|x\right|=2\right\}$$

Proof

Step	Hyp	Ref	Expression
1		umgrbi.x	$$⊢X\in V$$
2		umgrbi.y	$$⊢Y\in V$$
3		umgrbi.n	$$⊢X\ne Y$$
4		prssi	$$⊢\left(X\in V\wedge Y\in V\right)\to \left\{X,Y\right\}\subseteq V$$
5	1 2 4	mp2an	$$⊢\left\{X,Y\right\}\subseteq V$$
6		prex	$$⊢\left\{X,Y\right\}\in \mathrm{V}$$
7	6	elpw	$$⊢\left\{X,Y\right\}\in 𝒫V\leftrightarrow \left\{X,Y\right\}\subseteq V$$
8	5 7	mpbir	$$⊢\left\{X,Y\right\}\in 𝒫V$$
9		hashprg	$$⊢\left(X\in V\wedge Y\in V\right)\to \left(X\ne Y\leftrightarrow \left|\left\{X,Y\right\}\right|=2\right)$$
10	3 9	mpbii	$$⊢\left(X\in V\wedge Y\in V\right)\to \left|\left\{X,Y\right\}\right|=2$$
11	1 2 10	mp2an	$$⊢\left|\left\{X,Y\right\}\right|=2$$
12		fveqeq2	$$⊢x=\left\{X,Y\right\}\to \left(\left|x\right|=2\leftrightarrow \left|\left\{X,Y\right\}\right|=2\right)$$
13	12	elrab	$$⊢\left\{X,Y\right\}\in \left\{x\in 𝒫V|\left|x\right|=2\right\}\leftrightarrow \left(\left\{X,Y\right\}\in 𝒫V\wedge \left|\left\{X,Y\right\}\right|=2\right)$$
14	8 11 13	mpbir2an	$$⊢\left\{X,Y\right\}\in \left\{x\in 𝒫V|\left|x\right|=2\right\}$$