Theorem xrge0infss 31961

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 3977	. . . . . . 7 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (0[,]+∞))
2		0xr 11258	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
3		pnfxr 11265	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
4		iccgelb 13377	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
5	2, 3, 4	mp3an12 1452	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6		eliccxr 13409	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
7		xrlenlt 11276	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
8	2, 6, 7	sylancr 588	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
9	5, 8	mpbid 231	. . . . . . 7 ⊢ (𝑦 ∈ (0[,]+∞) → ¬ 𝑦 < 0)
10	1, 9	syl 17	. . . . . 6 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 < 0)
11	10	ralrimiva 3147	. . . . 5 ⊢ (𝐴 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0)
12	11	ad3antrrr 729	. . . 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 13404	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ*
14		ssralv 4050	. . . . . . . . . 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 3980	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 < 𝑦)
22	16, 17, 19, 20, 21	xrlelttrd 13136	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 < 𝑦)
23	22	ex 414	. . . . . . . . . . 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 3168	. . . . . . . . 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 713	. . . . . . 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 408	. . . . . 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 715	. . . . 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 651	. . . 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 13433	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
32		breq2 5152	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑦 < 𝑥 ↔ 𝑦 < 0))
33	32	notbid 318	. . . . . . . 8 ⊢ (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
34	33	ralbidv 3178	. . . . . . 7 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0))
35		breq1 5151	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
36	35	imbi1d 342	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
37	36	ralbidv 3178	. . . . . . 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 3613	. . . . 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 689	. . . 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 585	. . 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 486	. . . . 5 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 0 ≤ 𝑤)
44		elxrge0 13431	. . . . 5 ⊢ (𝑤 ∈ (0[,]+∞) ↔ (𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤))
45	42, 43, 44	sylanbrc 584	. . . 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 613	. . . . . . 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 482	. . . . . 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 408	. . . . 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 482	. . . 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 5152	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑤))
52	51	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
53	52	ralbidv 3178	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤))
54		breq1 5151	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 < 𝑦 ↔ 𝑤 < 𝑦))
55	54	imbi1d 342	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
56	55	ralbidv 3178	. . . . . 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 3613	. . . 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 585	. . 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 13129	. . . 4 ⊢ ((𝑤 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
63	60, 61, 62	syl2anc 585	. . 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 958	. 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 3990	. . . 4 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
66	13, 65	mpan2 690	. . 3 ⊢ (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67		xrinfmss 13286	. . 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 3163	1 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ⊆ wss 3948   class class class wbr 5148  (class class class)co 7406  0cc0 11107  +∞cpnf 11242  ℝ^*cxr 11244   < clt 11245   ≤ cle 11246  [,]cicc 13324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-icc 13328

This theorem is referenced by:  xrge0infssd  31962  infxrge0lb  31965  infxrge0glb  31966  infxrge0gelb  31967  omsf  33284  omssubaddlem  33287  omssubadd  33288