Theorem nosepne

Description: The value of two non-equal surreals at the first place they differ is different. (Contributed by Scott Fenton, 24-Nov-2021)

	Ref	Expression
Assertion	nosepne	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge A\ne B\right)\to A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		sltso	$$⊢{<}_s\mathrm{Or}\mathrm{No}$$
2		sotrine	$$⊢\left({<}_s\mathrm{Or}\mathrm{No}\wedge \left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\right)\to \left(A\ne B\leftrightarrow \left(A{<}_{s}B\vee B{<}_{s}A\right)\right)$$
3	1 2	mpan	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A\ne B\leftrightarrow \left(A{<}_{s}B\vee B{<}_{s}A\right)\right)$$
4		nosepnelem	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge A{<}_{s}B\right)\to A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)$$
5	4	3expia	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A{<}_{s}B\to A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\right)$$
6		nosepnelem	$$⊢\left(B\in \mathrm{No}\wedge A\in \mathrm{No}\wedge B{<}_{s}A\right)\to B\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)\ne A\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)$$
7		necom	$$⊢A\left(x\right)\ne B\left(x\right)\leftrightarrow B\left(x\right)\ne A\left(x\right)$$
8	7	rabbii	$$⊢\left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}=\left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}$$
9	8	inteqi	$$⊢\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}=\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}$$
10	9	fveq2i	$$⊢A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)=A\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)$$
11	9	fveq2i	$$⊢B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)=B\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)$$
12	10 11	neeq12i	$$⊢A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\leftrightarrow A\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)$$
13		necom	$$⊢A\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)\leftrightarrow B\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)\ne A\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)$$
14	12 13	bitri	$$⊢A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\leftrightarrow B\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)\ne A\left(\bigcap \left\{x\in \mathrm{On}|B\left(x\right)\ne A\left(x\right)\right\}\right)$$
15	6 14	sylibr	$$⊢\left(B\in \mathrm{No}\wedge A\in \mathrm{No}\wedge B{<}_{s}A\right)\to A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)$$
16	15	3expia	$$⊢\left(B\in \mathrm{No}\wedge A\in \mathrm{No}\right)\to \left(B{<}_{s}A\to A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\right)$$
17	16	ancoms	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(B{<}_{s}A\to A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\right)$$
18	5 17	jaod	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(\left(A{<}_{s}B\vee B{<}_{s}A\right)\to A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\right)$$
19	3 18	sylbid	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to \left(A\ne B\to A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\right)$$
20	19	3impia	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge A\ne B\right)\to A\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)\ne B\left(\bigcap \left\{x\in \mathrm{On}|A\left(x\right)\ne B\left(x\right)\right\}\right)$$