Theorem cfle

Description: Cofinality is bounded by its argument. Exercise 1 of TakeutiZaring p. 102. (Contributed by NM, 26-Apr-2004) (Revised by Mario Carneiro, 15-Sep-2013)

	Ref	Expression
Assertion	cfle	⊢ ( cf ‘ 𝐴 ) ⊆ 𝐴

Proof

Step	Hyp	Ref	Expression
1		cflecard	⊢ ( cf ‘ 𝐴 ) ⊆ ( card ‘ 𝐴 )
2		cardonle	⊢ ( 𝐴 ∈ On → ( card ‘ 𝐴 ) ⊆ 𝐴 )
3	1 2	sstrid	⊢ ( 𝐴 ∈ On → ( cf ‘ 𝐴 ) ⊆ 𝐴 )
4		cff	⊢ cf : On ⟶ On
5	4	fdmi	⊢ dom cf = On
6	5	eleq2i	⊢ ( 𝐴 ∈ dom cf ↔ 𝐴 ∈ On )
7		ndmfv	⊢ ( ¬ 𝐴 ∈ dom cf → ( cf ‘ 𝐴 ) = ∅ )
8	6 7	sylnbir	⊢ ( ¬ 𝐴 ∈ On → ( cf ‘ 𝐴 ) = ∅ )
9		0ss	⊢ ∅ ⊆ 𝐴
10	8 9	eqsstrdi	⊢ ( ¬ 𝐴 ∈ On → ( cf ‘ 𝐴 ) ⊆ 𝐴 )
11	3 10	pm2.61i	⊢ ( cf ‘ 𝐴 ) ⊆ 𝐴