Theorem oaun3

Description: Ordinal addition as a union of classes. (Contributed by RP, 13-Feb-2025)

	Ref	Expression
Assertion	oaun3	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A{+}_{𝑜}B=\bigcup \left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\},\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		oacl	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A{+}_{𝑜}B\in \mathrm{On}$$
2	1	difexd	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left(A{+}_{𝑜}B\right)\setminus A\in \mathrm{V}$$
3		uniprg	$$⊢\left(A\in \mathrm{On}\wedge \left(A{+}_{𝑜}B\right)\setminus A\in \mathrm{V}\right)\to \bigcup \left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}=A\cup \left(\left(A{+}_{𝑜}B\right)\setminus A\right)$$
4	2 3	syldan	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \bigcup \left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}=A\cup \left(\left(A{+}_{𝑜}B\right)\setminus A\right)$$
5		undif2	$$⊢A\cup \left(\left(A{+}_{𝑜}B\right)\setminus A\right)=A\cup \left(A{+}_{𝑜}B\right)$$
6		oaword1	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A\subseteq A{+}_{𝑜}B$$
7		ssequn1	$$⊢A\subseteq A{+}_{𝑜}B\leftrightarrow A\cup \left(A{+}_{𝑜}B\right)=A{+}_{𝑜}B$$
8	6 7	sylib	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A\cup \left(A{+}_{𝑜}B\right)=A{+}_{𝑜}B$$
9	5 8	eqtrid	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A\cup \left(\left(A{+}_{𝑜}B\right)\setminus A\right)=A{+}_{𝑜}B$$
10	4 9	eqtrd	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \bigcup \left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}=A{+}_{𝑜}B$$
11		oaun3lem4	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\in \mathrm{suc}\left(A{+}_{𝑜}B\right)$$
12		unisng	$$⊢\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\in \mathrm{suc}\left(A{+}_{𝑜}B\right)\to \bigcup \left\{\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}=\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}$$
13	11 12	syl	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \bigcup \left\{\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}=\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}$$
14	10 13	uneq12d	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \bigcup \left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}\cup \bigcup \left\{\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}=\left(A{+}_{𝑜}B\right)\cup \left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}$$
15		uniun	$$⊢\bigcup \left(\left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}\cup \left\{\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}\right)=\bigcup \left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}\cup \bigcup \left\{\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}$$
16		df-tp	$$⊢\left\{A,\left(A{+}_{𝑜}B\right)\setminus A,\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}=\left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}\cup \left\{\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}$$
17		rp-abid	$$⊢A=\left\{x|\exists a\in Ax=a\right\}$$
18	17	a1i	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A=\left\{x|\exists a\in Ax=a\right\}$$
19		oadif1	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left(A{+}_{𝑜}B\right)\setminus A=\left\{y|\exists b\in By=A{+}_{𝑜}b\right\}$$
20		eqidd	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}=\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}$$
21	18 19 20	tpeq123d	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left\{A,\left(A{+}_{𝑜}B\right)\setminus A,\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}=\left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\},\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}$$
22	16 21	eqtr3id	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}\cup \left\{\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}=\left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\},\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}$$
23	22	unieqd	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \bigcup \left(\left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}\cup \left\{\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}\right)=\bigcup \left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\},\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}$$
24	15 23	eqtr3id	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \bigcup \left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}\cup \bigcup \left\{\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}=\bigcup \left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\},\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}$$
25		oaun3lem2	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\subseteq A{+}_{𝑜}B$$
26		ssequn2	$$⊢\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\subseteq A{+}_{𝑜}B\leftrightarrow \left(A{+}_{𝑜}B\right)\cup \left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}=A{+}_{𝑜}B$$
27	25 26	sylib	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left(A{+}_{𝑜}B\right)\cup \left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}=A{+}_{𝑜}B$$
28	14 24 27	3eqtr3rd	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A{+}_{𝑜}B=\bigcup \left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\},\left\{z|\exists a\in A\exists b\in Bz=a{+}_{𝑜}b\right\}\right\}$$