Theorem pren2d

Description: A pair of two distinct sets is equinumerous to ordinal two. (Contributed by RP, 21-Oct-2023)

	Ref	Expression
Hypotheses	pren2d.a	$$⊢\phi \to A\in V$$
	pren2d.b	$$⊢\phi \to B\in W$$
	pren2d.aneb	$$⊢\phi \to A\ne B$$
Assertion	pren2d	$$⊢\phi \to \left\{A,B\right\}\approx 2_{𝑜}$$

Proof

Step	Hyp	Ref	Expression
1		pren2d.a	$$⊢\phi \to A\in V$$
2		pren2d.b	$$⊢\phi \to B\in W$$
3		pren2d.aneb	$$⊢\phi \to A\ne B$$
4	1	elexd	$$⊢\phi \to A\in \mathrm{V}$$
5	2	elexd	$$⊢\phi \to B\in \mathrm{V}$$
6		pren2	$$⊢\left\{A,B\right\}\approx 2_{𝑜}\leftrightarrow \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\wedge A\ne B\right)$$
7	4 5 3 6	syl3anbrc	$$⊢\phi \to \left\{A,B\right\}\approx 2_{𝑜}$$