Theorem noxp1o

Description: The Cartesian product of an ordinal and { 1o } is a surreal. (Contributed by Scott Fenton, 12-Jun-2011)

	Ref	Expression
Assertion	noxp1o	$$⊢A\in \mathrm{On}\to A\times \left\{1_{𝑜}\right\}\in \mathrm{No}$$

Proof

Step	Hyp	Ref	Expression
1		1oex	$$⊢1_{𝑜}\in \mathrm{V}$$
2	1	prid1	$$⊢1_{𝑜}\in \left\{1_{𝑜},2_{𝑜}\right\}$$
3	2	fconst6	$$⊢\left(A\times \left\{1_{𝑜}\right\}\right):A⟶\left\{1_{𝑜},2_{𝑜}\right\}$$
4	1	snnz	$$⊢\left\{1_{𝑜}\right\}\ne \varnothing$$
5		dmxp	$$⊢\left\{1_{𝑜}\right\}\ne \varnothing \to \mathrm{dom}\left(A\times \left\{1_{𝑜}\right\}\right)=A$$
6	4 5	ax-mp	$$⊢\mathrm{dom}\left(A\times \left\{1_{𝑜}\right\}\right)=A$$
7	6	feq2i	$$⊢\left(A\times \left\{1_{𝑜}\right\}\right):\mathrm{dom}\left(A\times \left\{1_{𝑜}\right\}\right)⟶\left\{1_{𝑜},2_{𝑜}\right\}\leftrightarrow \left(A\times \left\{1_{𝑜}\right\}\right):A⟶\left\{1_{𝑜},2_{𝑜}\right\}$$
8	3 7	mpbir	$$⊢\left(A\times \left\{1_{𝑜}\right\}\right):\mathrm{dom}\left(A\times \left\{1_{𝑜}\right\}\right)⟶\left\{1_{𝑜},2_{𝑜}\right\}$$
9	8	a1i	$$⊢A\in \mathrm{On}\to \left(A\times \left\{1_{𝑜}\right\}\right):\mathrm{dom}\left(A\times \left\{1_{𝑜}\right\}\right)⟶\left\{1_{𝑜},2_{𝑜}\right\}$$
10	6	eleq1i	$$⊢\mathrm{dom}\left(A\times \left\{1_{𝑜}\right\}\right)\in \mathrm{On}\leftrightarrow A\in \mathrm{On}$$
11	10	biimpri	$$⊢A\in \mathrm{On}\to \mathrm{dom}\left(A\times \left\{1_{𝑜}\right\}\right)\in \mathrm{On}$$
12		elno3	$$⊢A\times \left\{1_{𝑜}\right\}\in \mathrm{No}\leftrightarrow \left(\left(A\times \left\{1_{𝑜}\right\}\right):\mathrm{dom}\left(A\times \left\{1_{𝑜}\right\}\right)⟶\left\{1_{𝑜},2_{𝑜}\right\}\wedge \mathrm{dom}\left(A\times \left\{1_{𝑜}\right\}\right)\in \mathrm{On}\right)$$
13	9 11 12	sylanbrc	$$⊢A\in \mathrm{On}\to A\times \left\{1_{𝑜}\right\}\in \mathrm{No}$$