Theorem onprcr

Description: Relative version of onprc . (Contributed by Adrian Ducourtial, 15-Oct-2026)

	Ref	Expression
Hypotheses	onprcr.1	$$⊢A=\left\{x\in V|\mathrm{Ord}x\right\}$$
	onprcr.2	$$⊢\mathrm{Tr}A$$
Assertion	onprcr	$$⊢\neg A\in V$$

Proof

Step	Hyp	Ref	Expression
1		onprcr.1	$$⊢A=\left\{x\in V|\mathrm{Ord}x\right\}$$
2		onprcr.2	$$⊢\mathrm{Tr}A$$
3		rabssab	$$⊢\left\{x\in V|\mathrm{Ord}x\right\}\subseteq \left\{x|\mathrm{Ord}x\right\}$$
4		df-on	$$⊢\mathrm{On}=\left\{x|\mathrm{Ord}x\right\}$$
5	3 1 4	3sstr4i	$$⊢A\subseteq \mathrm{On}$$
6		epweon	$$⊢\mathrm{E}\mathrm{We}\mathrm{On}$$
7		wess	$$⊢A\subseteq \mathrm{On}\to \left(\mathrm{E}\mathrm{We}\mathrm{On}\to \mathrm{E}\mathrm{We}A\right)$$
8	5 6 7	mp2	$$⊢\mathrm{E}\mathrm{We}A$$
9		df-ord	$$⊢\mathrm{Ord}A\leftrightarrow \left(\mathrm{Tr}A\wedge \mathrm{E}\mathrm{We}A\right)$$
10	2 8 9	mpbir2an	$$⊢\mathrm{Ord}A$$
11		ordirr	$$⊢\mathrm{Ord}A\to \neg A\in A$$
12	10 11	ax-mp	$$⊢\neg A\in A$$
13		ordeq	$$⊢x=A\to \left(\mathrm{Ord}x\leftrightarrow \mathrm{Ord}A\right)$$
14	13	elrab3	$$⊢A\in V\to \left(A\in \left\{x\in V|\mathrm{Ord}x\right\}\leftrightarrow \mathrm{Ord}A\right)$$
15	10 14	mpbiri	$$⊢A\in V\to A\in \left\{x\in V|\mathrm{Ord}x\right\}$$
16	15 1	eleqtrrdi	$$⊢A\in V\to A\in A$$
17	12 16	mto	$$⊢\neg A\in V$$