Theorem naddov3

Description: Alternate expression for natural addition. (Contributed by Scott Fenton, 20-Jan-2025)

	Ref	Expression
Assertion	naddov3	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A+B=\bigcap \left\{x\in \mathrm{On}|+\left[\left\{A\right\}\times B\right]\cup +\left[A\times \left\{B\right\}\right]\subseteq x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		naddov	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A+B=\bigcap \left\{x\in \mathrm{On}|\left(+\left[\left\{A\right\}\times B\right]\subseteq x\wedge +\left[A\times \left\{B\right\}\right]\subseteq x\right)\right\}$$
2		unss	$$⊢\left(+\left[\left\{A\right\}\times B\right]\subseteq x\wedge +\left[A\times \left\{B\right\}\right]\subseteq x\right)\leftrightarrow +\left[\left\{A\right\}\times B\right]\cup +\left[A\times \left\{B\right\}\right]\subseteq x$$
3	2	rabbii	$$⊢\left\{x\in \mathrm{On}|\left(+\left[\left\{A\right\}\times B\right]\subseteq x\wedge +\left[A\times \left\{B\right\}\right]\subseteq x\right)\right\}=\left\{x\in \mathrm{On}|+\left[\left\{A\right\}\times B\right]\cup +\left[A\times \left\{B\right\}\right]\subseteq x\right\}$$
4	3	inteqi	$$⊢\bigcap \left\{x\in \mathrm{On}|\left(+\left[\left\{A\right\}\times B\right]\subseteq x\wedge +\left[A\times \left\{B\right\}\right]\subseteq x\right)\right\}=\bigcap \left\{x\in \mathrm{On}|+\left[\left\{A\right\}\times B\right]\cup +\left[A\times \left\{B\right\}\right]\subseteq x\right\}$$
5	1 4	eqtrdi	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A+B=\bigcap \left\{x\in \mathrm{On}|+\left[\left\{A\right\}\times B\right]\cup +\left[A\times \left\{B\right\}\right]\subseteq x\right\}$$