Theorem alephsson

Description: The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003)

	Ref	Expression
Assertion	alephsson	⊢ ran ℵ ⊆ On

Proof

Step	Hyp	Ref	Expression
1		isinfcard	⊢ ( ( ω ⊆ 𝑥 ∧ ( card ‘ 𝑥 ) = 𝑥 ) ↔ 𝑥 ∈ ran ℵ )
2		cardon	⊢ ( card ‘ 𝑥 ) ∈ On
3		eleq1	⊢ ( ( card ‘ 𝑥 ) = 𝑥 → ( ( card ‘ 𝑥 ) ∈ On ↔ 𝑥 ∈ On ) )
4	2 3	mpbii	⊢ ( ( card ‘ 𝑥 ) = 𝑥 → 𝑥 ∈ On )
5	4	adantl	⊢ ( ( ω ⊆ 𝑥 ∧ ( card ‘ 𝑥 ) = 𝑥 ) → 𝑥 ∈ On )
6	1 5	sylbir	⊢ ( 𝑥 ∈ ran ℵ → 𝑥 ∈ On )
7	6	ssriv	⊢ ran ℵ ⊆ On