Theorem n0snor2el

Description: A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020)

	Ref	Expression
Assertion	n0snor2el	$$⊢A\ne \varnothing \to \left(\exists x\in A\exists y\in Ax\ne y\vee \exists zA=\left\{z\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		issn	$$⊢\exists w\in A\forall y\in Aw=y\to \exists zA=\left\{z\right\}$$
2	1	olcd	$$⊢\exists w\in A\forall y\in Aw=y\to \left(\exists x\in A\exists y\in Ax\ne y\vee \exists zA=\left\{z\right\}\right)$$
3	2	a1d	$$⊢\exists w\in A\forall y\in Aw=y\to \left(A\ne \varnothing \to \left(\exists x\in A\exists y\in Ax\ne y\vee \exists zA=\left\{z\right\}\right)\right)$$
4		df-ne	$$⊢w\ne y\leftrightarrow \neg w=y$$
5	4	rexbii	$$⊢\exists y\in Aw\ne y\leftrightarrow \exists y\in A\neg w=y$$
6		rexnal	$$⊢\exists y\in A\neg w=y\leftrightarrow \neg \forall y\in Aw=y$$
7	5 6	bitri	$$⊢\exists y\in Aw\ne y\leftrightarrow \neg \forall y\in Aw=y$$
8	7	ralbii	$$⊢\forall w\in A\exists y\in Aw\ne y\leftrightarrow \forall w\in A\neg \forall y\in Aw=y$$
9		ralnex	$$⊢\forall w\in A\neg \forall y\in Aw=y\leftrightarrow \neg \exists w\in A\forall y\in Aw=y$$
10	8 9	bitri	$$⊢\forall w\in A\exists y\in Aw\ne y\leftrightarrow \neg \exists w\in A\forall y\in Aw=y$$
11		neeq1	$$⊢w=x\to \left(w\ne y\leftrightarrow x\ne y\right)$$
12	11	rexbidv	$$⊢w=x\to \left(\exists y\in Aw\ne y\leftrightarrow \exists y\in Ax\ne y\right)$$
13	12	rspccva	$$⊢\left(\forall w\in A\exists y\in Aw\ne y\wedge x\in A\right)\to \exists y\in Ax\ne y$$
14	13	reximdva0	$$⊢\left(\forall w\in A\exists y\in Aw\ne y\wedge A\ne \varnothing \right)\to \exists x\in A\exists y\in Ax\ne y$$
15	14	orcd	$$⊢\left(\forall w\in A\exists y\in Aw\ne y\wedge A\ne \varnothing \right)\to \left(\exists x\in A\exists y\in Ax\ne y\vee \exists zA=\left\{z\right\}\right)$$
16	15	ex	$$⊢\forall w\in A\exists y\in Aw\ne y\to \left(A\ne \varnothing \to \left(\exists x\in A\exists y\in Ax\ne y\vee \exists zA=\left\{z\right\}\right)\right)$$
17	10 16	sylbir	$$⊢\neg \exists w\in A\forall y\in Aw=y\to \left(A\ne \varnothing \to \left(\exists x\in A\exists y\in Ax\ne y\vee \exists zA=\left\{z\right\}\right)\right)$$
18	3 17	pm2.61i	$$⊢A\ne \varnothing \to \left(\exists x\in A\exists y\in Ax\ne y\vee \exists zA=\left\{z\right\}\right)$$