Theorem naddov

Description: The value of natural addition. (Contributed by Scott Fenton, 26-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		naddcllem	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left(A+B\in \mathrm{On}\wedge A+B=\bigcap \left\{x\in \mathrm{On}|\left(+\left[\left\{A\right\}\times B\right]\subseteq x\wedge +\left[A\times \left\{B\right\}\right]\subseteq x\right)\right\}\right)$$
2	1	simprd	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to A+B=\bigcap \left\{x\in \mathrm{On}|\left(+\left[\left\{A\right\}\times B\right]\subseteq x\wedge +\left[A\times \left\{B\right\}\right]\subseteq x\right)\right\}$$