Theorem hsmex2

Description: The set of hereditary size-limited sets, assuming ax-reg . (Contributed by Stefan O'Rear, 11-Feb-2015)

	Ref	Expression
Assertion	hsmex2	$$⊢X\in V\to \left\{s|\forall x\in \mathrm{TC}\left(\left\{s\right\}\right)x\preccurlyeq X\right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		unir1	$$⊢\bigcup R_1\left[\mathrm{On}\right]=\mathrm{V}$$
2	1	rabeqi	$$⊢\left\{s\in \bigcup R_1\left[\mathrm{On}\right]|\forall x\in \mathrm{TC}\left(\left\{s\right\}\right)x\preccurlyeq X\right\}=\left\{s\in \mathrm{V}|\forall x\in \mathrm{TC}\left(\left\{s\right\}\right)x\preccurlyeq X\right\}$$
3		rabab	$$⊢\left\{s\in \mathrm{V}|\forall x\in \mathrm{TC}\left(\left\{s\right\}\right)x\preccurlyeq X\right\}=\left\{s|\forall x\in \mathrm{TC}\left(\left\{s\right\}\right)x\preccurlyeq X\right\}$$
4	2 3	eqtr2i	$$⊢\left\{s|\forall x\in \mathrm{TC}\left(\left\{s\right\}\right)x\preccurlyeq X\right\}=\left\{s\in \bigcup R_1\left[\mathrm{On}\right]|\forall x\in \mathrm{TC}\left(\left\{s\right\}\right)x\preccurlyeq X\right\}$$
5		hsmex	$$⊢X\in V\to \left\{s\in \bigcup R_1\left[\mathrm{On}\right]|\forall x\in \mathrm{TC}\left(\left\{s\right\}\right)x\preccurlyeq X\right\}\in \mathrm{V}$$
6	4 5	eqeltrid	$$⊢X\in V\to \left\{s|\forall x\in \mathrm{TC}\left(\left\{s\right\}\right)x\preccurlyeq X\right\}\in \mathrm{V}$$