Theorem onwf

Description: The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	onwf	$$⊢\mathrm{On}\subseteq \bigcup R_1\left[\mathrm{On}\right]$$

Proof

Step	Hyp	Ref	Expression
1		r1fnon	$$⊢R_{1}Fn\mathrm{On}$$
2	1	fndmi	$$⊢\mathrm{dom}{R}_1=\mathrm{On}$$
3		rankonidlem	$$⊢x\in \mathrm{dom}{R}_1\to \left(x\in \bigcup R_1\left[\mathrm{On}\right]\wedge \mathrm{rank}\left(x\right)=x\right)$$
4	3	simpld	$$⊢x\in \mathrm{dom}{R}_1\to x\in \bigcup R_1\left[\mathrm{On}\right]$$
5	4	ssriv	$$⊢\mathrm{dom}{R}_1\subseteq \bigcup R_1\left[\mathrm{On}\right]$$
6	2 5	eqsstrri	$$⊢\mathrm{On}\subseteq \bigcup R_1\left[\mathrm{On}\right]$$