Theorem onsdom

Description: Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009) (Proof shortened by Mario Carneiro, 15-May-2015)

	Ref	Expression
Assertion	onsdom	$$⊢A\in \mathrm{dom}\mathrm{card}\to \exists x\in \mathrm{On}A\prec x$$

Proof

Step	Hyp	Ref	Expression
1		harcl	$$⊢\mathrm{har}\left(A\right)\in \mathrm{On}$$
2		harsdom	$$⊢A\in \mathrm{dom}\mathrm{card}\to A\prec \mathrm{har}\left(A\right)$$
3		breq2	$$⊢x=\mathrm{har}\left(A\right)\to \left(A\prec x\leftrightarrow A\prec \mathrm{har}\left(A\right)\right)$$
4	3	rspcev	$$⊢\left(\mathrm{har}\left(A\right)\in \mathrm{On}\wedge A\prec \mathrm{har}\left(A\right)\right)\to \exists x\in \mathrm{On}A\prec x$$
5	1 2 4	sylancr	$$⊢A\in \mathrm{dom}\mathrm{card}\to \exists x\in \mathrm{On}A\prec x$$