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 ∈ V
2	1	canth2	⊢ ℕ ≺ 𝒫 ℕ
3		rpnnen	⊢ ℝ ≈ 𝒫 ℕ
4	3	ensymi	⊢ 𝒫 ℕ ≈ ℝ
5		sdomentr	⊢ ( ( ℕ ≺ 𝒫 ℕ ∧ 𝒫 ℕ ≈ ℝ ) → ℕ ≺ ℝ )
6	2 4 5	mp2an	⊢ ℕ ≺ ℝ