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Metamath Proof Explorer

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	$⊢ ℕ ≺ ℝ$

Proof

Step	Hyp	Ref	Expression
1		nnex	$⊢ ℕ \in V$
2	1	canth2	$⊢ ℕ ≺ 𝒫 ℕ$
3		rpnnen	$⊢ ℝ \approx 𝒫 ℕ$
4	3	ensymi	$⊢ 𝒫 ℕ \approx ℝ$
5		sdomentr	$⊢ (ℕ ≺ 𝒫 ℕ \land 𝒫 ℕ \approx ℝ) \to ℕ ≺ ℝ$
6	2 4 5	mp2an	$⊢ ℕ ≺ ℝ$