Theorem rencldnfi

Description: A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem using infima; this theorem removes the requirement that A be nonempty. (Contributed by Stefan O'Rear, 19-Oct-2014)

	Ref	Expression
Assertion	rencldnfi	$$⊢\left(\left(A\subseteq ℝ\wedge B\in ℝ\wedge \neg B\in A\right)\wedge \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\right)\to \neg A\in \mathrm{Fin}$$

Proof

Step	Hyp	Ref	Expression
1		simpl1	$$⊢\left(\left(A\subseteq ℝ\wedge B\in ℝ\wedge \neg B\in A\right)\wedge \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\right)\to A\subseteq ℝ$$
2		simpl2	$$⊢\left(\left(A\subseteq ℝ\wedge B\in ℝ\wedge \neg B\in A\right)\wedge \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\right)\to B\in ℝ$$
3		rexn0	$$⊢\exists y\in A\left|y-B\right|<x\to A\ne \varnothing$$
4	3	ralimi	$$⊢\forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\to \forall x\in {ℝ}^{+}A\ne \varnothing$$
5		1rp	$$⊢1\in {ℝ}^{+}$$
6		ne0i	$$⊢1\in {ℝ}^{+}\to {ℝ}^{+}\ne \varnothing$$
7		r19.3rzv	$$⊢{ℝ}^{+}\ne \varnothing \to \left(A\ne \varnothing \leftrightarrow \forall x\in {ℝ}^{+}A\ne \varnothing \right)$$
8	5 6 7	mp2b	$$⊢A\ne \varnothing \leftrightarrow \forall x\in {ℝ}^{+}A\ne \varnothing$$
9	4 8	sylibr	$$⊢\forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\to A\ne \varnothing$$
10	9	adantl	$$⊢\left(\left(A\subseteq ℝ\wedge B\in ℝ\wedge \neg B\in A\right)\wedge \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\right)\to A\ne \varnothing$$
11		simpl3	$$⊢\left(\left(A\subseteq ℝ\wedge B\in ℝ\wedge \neg B\in A\right)\wedge \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\right)\to \neg B\in A$$
12	10 11	jca	$$⊢\left(\left(A\subseteq ℝ\wedge B\in ℝ\wedge \neg B\in A\right)\wedge \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\right)\to \left(A\ne \varnothing \wedge \neg B\in A\right)$$
13		simpr	$$⊢\left(\left(A\subseteq ℝ\wedge B\in ℝ\wedge \neg B\in A\right)\wedge \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\right)\to \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x$$
14		rencldnfilem	$$⊢\left(\left(A\subseteq ℝ\wedge B\in ℝ\wedge \left(A\ne \varnothing \wedge \neg B\in A\right)\right)\wedge \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\right)\to \neg A\in \mathrm{Fin}$$
15	1 2 12 13 14	syl31anc	$$⊢\left(\left(A\subseteq ℝ\wedge B\in ℝ\wedge \neg B\in A\right)\wedge \forall x\in {ℝ}^{+}\exists y\in A\left|y-B\right|<x\right)\to \neg A\in \mathrm{Fin}$$