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	$$⊢\mathrm{cf}\left(A\right)\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		cflecard	$$⊢\mathrm{cf}\left(A\right)\subseteq \mathrm{card}\left(A\right)$$
2		cardonle	$$⊢A\in \mathrm{On}\to \mathrm{card}\left(A\right)\subseteq A$$
3	1 2	sstrid	$$⊢A\in \mathrm{On}\to \mathrm{cf}\left(A\right)\subseteq A$$
4		cff	$$⊢\mathrm{cf}:\mathrm{On}⟶\mathrm{On}$$
5	4	fdmi	$$⊢\mathrm{dom}\mathrm{cf}=\mathrm{On}$$
6	5	eleq2i	$$⊢A\in \mathrm{dom}\mathrm{cf}\leftrightarrow A\in \mathrm{On}$$
7		ndmfv	$$⊢\neg A\in \mathrm{dom}\mathrm{cf}\to \mathrm{cf}\left(A\right)=\varnothing$$
8	6 7	sylnbir	$$⊢\neg A\in \mathrm{On}\to \mathrm{cf}\left(A\right)=\varnothing$$
9		0ss	$$⊢\varnothing \subseteq A$$
10	8 9	eqsstrdi	$$⊢\neg A\in \mathrm{On}\to \mathrm{cf}\left(A\right)\subseteq A$$
11	3 10	pm2.61i	$$⊢\mathrm{cf}\left(A\right)\subseteq A$$