Theorem ispointN

Description: The predicate "is a point". (Contributed by NM, 2-Oct-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ispoint.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	ispoint.p	$$⊢P=\mathrm{Points}\left(K\right)$$
Assertion	ispointN	$$⊢K\in D\to \left(X\in P\leftrightarrow \exists a\in AX=\left\{a\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		ispoint.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
2		ispoint.p	$$⊢P=\mathrm{Points}\left(K\right)$$
3	1 2	pointsetN	$$⊢K\in D\to P=\left\{x|\exists a\in Ax=\left\{a\right\}\right\}$$
4	3	eleq2d	$$⊢K\in D\to \left(X\in P\leftrightarrow X\in \left\{x|\exists a\in Ax=\left\{a\right\}\right\}\right)$$
5		snex	$$⊢\left\{a\right\}\in \mathrm{V}$$
6		eleq1	$$⊢X=\left\{a\right\}\to \left(X\in \mathrm{V}\leftrightarrow \left\{a\right\}\in \mathrm{V}\right)$$
7	5 6	mpbiri	$$⊢X=\left\{a\right\}\to X\in \mathrm{V}$$
8	7	rexlimivw	$$⊢\exists a\in AX=\left\{a\right\}\to X\in \mathrm{V}$$
9		eqeq1	$$⊢x=X\to \left(x=\left\{a\right\}\leftrightarrow X=\left\{a\right\}\right)$$
10	9	rexbidv	$$⊢x=X\to \left(\exists a\in Ax=\left\{a\right\}\leftrightarrow \exists a\in AX=\left\{a\right\}\right)$$
11	8 10	elab3	$$⊢X\in \left\{x|\exists a\in Ax=\left\{a\right\}\right\}\leftrightarrow \exists a\in AX=\left\{a\right\}$$
12	4 11	bitrdi	$$⊢K\in D\to \left(X\in P\leftrightarrow \exists a\in AX=\left\{a\right\}\right)$$