Theorem hashxrcl

Description: Extended real closure of the # function. (Contributed by Mario Carneiro, 22-Apr-2015)

	Ref	Expression
Assertion	hashxrcl	$$⊢A\in V\to \left|A\right|\in {ℝ}^{*}$$

Proof

Step	Hyp	Ref	Expression
1		nn0ssre	$$⊢{\mathbb{N}}_0\subseteq ℝ$$
2		ressxr	$$⊢ℝ\subseteq {ℝ}^{*}$$
3	1 2	sstri	$$⊢{\mathbb{N}}_0\subseteq {ℝ}^{*}$$
4		pnfxr	$$⊢\mathrm{+\infty }\in {ℝ}^{*}$$
5		snssi	$$⊢\mathrm{+\infty }\in {ℝ}^{*}\to \left\{\mathrm{+\infty }\right\}\subseteq {ℝ}^{*}$$
6	4 5	ax-mp	$$⊢\left\{\mathrm{+\infty }\right\}\subseteq {ℝ}^{*}$$
7	3 6	unssi	$$⊢{\mathbb{N}}_0\cup \left\{\mathrm{+\infty }\right\}\subseteq {ℝ}^{*}$$
8		elex	$$⊢A\in V\to A\in \mathrm{V}$$
9		hashf	$$⊢\left|.\right|:\mathrm{V}⟶{\mathbb{N}}_0\cup \left\{\mathrm{+\infty }\right\}$$
10	9	ffvelrni	$$⊢A\in \mathrm{V}\to \left|A\right|\in \left({\mathbb{N}}_0\cup \left\{\mathrm{+\infty }\right\}\right)$$
11	8 10	syl	$$⊢A\in V\to \left|A\right|\in \left({\mathbb{N}}_0\cup \left\{\mathrm{+\infty }\right\}\right)$$
12	7 11	sseldi	$$⊢A\in V\to \left|A\right|\in {ℝ}^{*}$$