Theorem rucALT

Description: Alternate proof of ruc . This proof is a simple corollary of rpnnen , which determines the exact cardinality of the reals. For an alternate proof discussed at mmcomplex.html#uncountable , see ruc . (Contributed by NM, 13-Oct-2004) (Revised by Mario Carneiro, 13-May-2013) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	rucALT	$$⊢\mathbb{N}\prec ℝ$$

Proof

Step	Hyp	Ref	Expression
1		nnex	$$⊢\mathbb{N}\in \mathrm{V}$$
2	1	canth2	$$⊢\mathbb{N}\prec 𝒫\mathbb{N}$$
3		rpnnen	$$⊢ℝ\approx 𝒫\mathbb{N}$$
4	3	ensymi	$$⊢𝒫\mathbb{N}\approx ℝ$$
5		sdomentr	$$⊢\left(\mathbb{N}\prec 𝒫\mathbb{N}\wedge 𝒫\mathbb{N}\approx ℝ\right)\to \mathbb{N}\prec ℝ$$
6	2 4 5	mp2an	$$⊢\mathbb{N}\prec ℝ$$