Theorem xrge0infss 32840

Description: Any subset of nonnegative extended reals has an infimum. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)

Assertion

Ref	Expression
xrge0infss	⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem xrge0infss

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel2 3928	. . . . . . 7 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (0[,]+∞))
2		0xr 11179	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
3		pnfxr 11186	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
4		iccgelb 13318	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
5	2, 3, 4	mp3an12 1453	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6		eliccxr 13351	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
7		xrlenlt 11197	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
8	2, 6, 7	sylancr 587	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
9	5, 8	mpbid 232	. . . . . . 7 ⊢ (𝑦 ∈ (0[,]+∞) → ¬ 𝑦 < 0)
10	1, 9	syl 17	. . . . . 6 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 < 0)
11	10	ralrimiva 3128	. . . . 5 ⊢ (𝐴 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0)
12	11	ad3antrrr 730	. . . 4 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0)
13		iccssxr 13346	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ*
14		ssralv 4002	. . . . . . . . . 10 ⊢ ((0[,]+∞) ⊆ ℝ* → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
16		simplll 774	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 ∈ ℝ*)
17	2	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 ∈ ℝ*)
18		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ (0[,]+∞))
19	13, 18	sselid 3931	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ*)
20		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 ≤ 0)
21		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 < 𝑦)
22	16, 17, 19, 20, 21	xrlelttrd 13074	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 < 𝑦)
23	22	ex 412	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) → (0 < 𝑦 → 𝑤 < 𝑦))
24	23	imim1d 82	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) → ((𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → (0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
25	24	ralimdva 3148	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) → (∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
26	15, 25	syl5 34	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
27	26	adantll 714	. . . . . . 7 ⊢ (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
28	27	imp 406	. . . . . 6 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
29	28	adantrl 716	. . . . 5 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ 𝑤 ≤ 0) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
30	29	an32s 652	. . . 4 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
31		0e0iccpnf 13375	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
32		breq2 5102	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑦 < 𝑥 ↔ 𝑦 < 0))
33	32	notbid 318	. . . . . . . 8 ⊢ (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
34	33	ralbidv 3159	. . . . . . 7 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0))
35		breq1 5101	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
36	35	imbi1d 341	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
37	36	ralbidv 3159	. . . . . . 7 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
38	34, 37	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 0 → ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
39	38	rspcev 3576	. . . . 5 ⊢ ((0 ∈ (0[,]+∞) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
40	31, 39	mpan 690	. . . 4 ⊢ ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0 ∧ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
41	12, 30, 40	syl2anc 584	. . 3 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
42		simpllr 775	. . . . 5 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 𝑤 ∈ ℝ*)
43		simpr 484	. . . . 5 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 0 ≤ 𝑤)
44		elxrge0 13373	. . . . 5 ⊢ (𝑤 ∈ (0[,]+∞) ↔ (𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤))
45	42, 43, 44	sylanbrc 583	. . . 4 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 𝑤 ∈ (0[,]+∞))
46	15	a1i 11	. . . . . . . 8 ⊢ (𝐴 ⊆ (0[,]+∞) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
47	46	anim2d 612	. . . . . . 7 ⊢ (𝐴 ⊆ (0[,]+∞) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
48	47	adantr 480	. . . . . 6 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) → ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
49	48	imp 406	. . . . 5 ⊢ (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
50	49	adantr 480	. . . 4 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
51		breq2 5102	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑤))
52	51	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
53	52	ralbidv 3159	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤))
54		breq1 5101	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 < 𝑦 ↔ 𝑤 < 𝑦))
55	54	imbi1d 341	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
56	55	ralbidv 3159	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
57	53, 56	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑤 → ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
58	57	rspcev 3576	. . . 4 ⊢ ((𝑤 ∈ (0[,]+∞) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
59	45, 50, 58	syl2anc 584	. . 3 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
60		simplr 768	. . . 4 ⊢ (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → 𝑤 ∈ ℝ*)
61	2	a1i 11	. . . 4 ⊢ (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → 0 ∈ ℝ*)
62		xrletri 13067	. . . 4 ⊢ ((𝑤 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
63	60, 61, 62	syl2anc 584	. . 3 ⊢ (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
64	41, 59, 63	mpjaodan 960	. 2 ⊢ (((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
65		sstr 3942	. . . 4 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
66	13, 65	mpan2 691	. . 3 ⊢ (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67		xrinfmss 13225	. . 3 ⊢ (𝐴 ⊆ ℝ* → ∃𝑤 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
68	66, 67	syl 17	. 2 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑤 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
69	64, 68	r19.29a 3144	1 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901   class class class wbr 5098  (class class class)co 7358  0cc0 11026  +∞cpnf 11163  ℝ^*cxr 11165   < clt 11166   ≤ cle 11167  [,]cicc 13264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-icc 13268

This theorem is referenced by:  xrge0infssd  32841  infxrge0lb  32844  infxrge0glb  32845  infxrge0gelb  32846  omsf  34453  omssubaddlem  34456  omssubadd  34457