Theorem fiminre

Description: A nonempty finite set of real numbers has a minimum. Analogous to fimaxre . (Contributed by AV, 9-Aug-2020) (Proof shortened by Steven Nguyen, 3-Jun-2023)

	Ref	Expression
Assertion	fiminre	$$⊢\left(A\subseteq ℝ\wedge A\in \mathrm{Fin}\wedge A\ne \varnothing \right)\to \exists x\in A\forall y\in Ax\le y$$

Proof

Step	Hyp	Ref	Expression
1		ltso	$$⊢<\mathrm{Or}ℝ$$
2		soss	$$⊢A\subseteq ℝ\to \left(<\mathrm{Or}ℝ\to <\mathrm{Or}A\right)$$
3	1 2	mpi	$$⊢A\subseteq ℝ\to <\mathrm{Or}A$$
4		fiming	$$⊢\left(<\mathrm{Or}A\wedge A\in \mathrm{Fin}\wedge A\ne \varnothing \right)\to \exists x\in A\forall y\in A\left(x\ne y\to x<y\right)$$
5	3 4	syl3an1	$$⊢\left(A\subseteq ℝ\wedge A\in \mathrm{Fin}\wedge A\ne \varnothing \right)\to \exists x\in A\forall y\in A\left(x\ne y\to x<y\right)$$
6		ssel2	$$⊢\left(A\subseteq ℝ\wedge x\in A\right)\to x\in ℝ$$
7	6	adantr	$$⊢\left(\left(A\subseteq ℝ\wedge x\in A\right)\wedge y\in A\right)\to x\in ℝ$$
8		ssel2	$$⊢\left(A\subseteq ℝ\wedge y\in A\right)\to y\in ℝ$$
9	8	adantlr	$$⊢\left(\left(A\subseteq ℝ\wedge x\in A\right)\wedge y\in A\right)\to y\in ℝ$$
10	7 9	leloed	$$⊢\left(\left(A\subseteq ℝ\wedge x\in A\right)\wedge y\in A\right)\to \left(x\le y\leftrightarrow \left(x<y\vee x=y\right)\right)$$
11		orcom	$$⊢\left(x=y\vee x<y\right)\leftrightarrow \left(x<y\vee x=y\right)$$
12	11	a1i	$$⊢\left(\left(A\subseteq ℝ\wedge x\in A\right)\wedge y\in A\right)\to \left(\left(x=y\vee x<y\right)\leftrightarrow \left(x<y\vee x=y\right)\right)$$
13		neor	$$⊢\left(x=y\vee x<y\right)\leftrightarrow \left(x\ne y\to x<y\right)$$
14	13	a1i	$$⊢\left(\left(A\subseteq ℝ\wedge x\in A\right)\wedge y\in A\right)\to \left(\left(x=y\vee x<y\right)\leftrightarrow \left(x\ne y\to x<y\right)\right)$$
15	10 12 14	3bitr2d	$$⊢\left(\left(A\subseteq ℝ\wedge x\in A\right)\wedge y\in A\right)\to \left(x\le y\leftrightarrow \left(x\ne y\to x<y\right)\right)$$
16	15	biimprd	$$⊢\left(\left(A\subseteq ℝ\wedge x\in A\right)\wedge y\in A\right)\to \left(\left(x\ne y\to x<y\right)\to x\le y\right)$$
17	16	ralimdva	$$⊢\left(A\subseteq ℝ\wedge x\in A\right)\to \left(\forall y\in A\left(x\ne y\to x<y\right)\to \forall y\in Ax\le y\right)$$
18	17	reximdva	$$⊢A\subseteq ℝ\to \left(\exists x\in A\forall y\in A\left(x\ne y\to x<y\right)\to \exists x\in A\forall y\in Ax\le y\right)$$
19	18	3ad2ant1	$$⊢\left(A\subseteq ℝ\wedge A\in \mathrm{Fin}\wedge A\ne \varnothing \right)\to \left(\exists x\in A\forall y\in A\left(x\ne y\to x<y\right)\to \exists x\in A\forall y\in Ax\le y\right)$$
20	5 19	mpd	$$⊢\left(A\subseteq ℝ\wedge A\in \mathrm{Fin}\wedge A\ne \varnothing \right)\to \exists x\in A\forall y\in Ax\le y$$