Theorem oaun2

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

	Ref	Expression
Assertion	oaun2	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A{+}_{𝑜}B=\bigcup \left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\}\right\}$$

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		rp-abid	$$⊢A=\left\{x|\exists a\in Ax=a\right\}$$
6	5	a1i	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A=\left\{x|\exists a\in Ax=a\right\}$$
7		oadif1	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left(A{+}_{𝑜}B\right)\setminus A=\left\{y|\exists b\in By=A{+}_{𝑜}b\right\}$$
8	6 7	preq12d	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}=\left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\}\right\}$$
9	8	unieqd	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \bigcup \left\{A,\left(A{+}_{𝑜}B\right)\setminus A\right\}=\bigcup \left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\}\right\}$$
10		undif2	$$⊢A\cup \left(\left(A{+}_{𝑜}B\right)\setminus A\right)=A\cup \left(A{+}_{𝑜}B\right)$$
11		oaword1	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A\subseteq A{+}_{𝑜}B$$
12		ssequn1	$$⊢A\subseteq A{+}_{𝑜}B\leftrightarrow A\cup \left(A{+}_{𝑜}B\right)=A{+}_{𝑜}B$$
13	11 12	sylib	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A\cup \left(A{+}_{𝑜}B\right)=A{+}_{𝑜}B$$
14	10 13	eqtrid	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A\cup \left(\left(A{+}_{𝑜}B\right)\setminus A\right)=A{+}_{𝑜}B$$
15	4 9 14	3eqtr3rd	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A{+}_{𝑜}B=\bigcup \left\{\left\{x|\exists a\in Ax=a\right\},\left\{y|\exists b\in By=A{+}_{𝑜}b\right\}\right\}$$