Theorem xrge0infss 33017

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 3934	. . . . . . 7 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (0[,]+∞))
2		0xr 11244	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
3		pnfxr 11251	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
4		iccgelb 13420	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
5	2, 3, 4	mp3an12 1475	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6		eliccxr 13453	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
7		xrlenlt 11262	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
8	2, 6, 7	sylancr 598	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
9	5, 8	mpbid 235	. . . . . . 7 ⊢ (𝑦 ∈ (0[,]+∞) → ¬ 𝑦 < 0)
10	1, 9	syl 18	. . . . . 6 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 < 0)
11	10	ralrimiva 3157	. . . . 5 ⊢ (𝐴 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0)
12	11	ad3antrrr 742	. . . 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 13448	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ*
14		ssralv 4008	. . . . . . . . . 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 786	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ (0[,]+∞))
19	13, 18	sselid 3937	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ*)
20		simpllr 787	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 ≤ 0)
21		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 < 𝑦)
22	16, 17, 19, 20, 21	xrlelttrd 13176	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 < 𝑦)
23	22	ex 417	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) → (0 < 𝑦 → 𝑤 < 𝑦))
24	23	imim1d 83	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) → ((𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → (0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
25	24	ralimdva 3177	. . . . . . . . 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 35	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
27	26	adantll 726	. . . . . . 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 411	. . . . . 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 728	. . . . 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 664	. . . 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 13477	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
32		breq2 5109	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑦 < 𝑥 ↔ 𝑦 < 0))
33	32	notbid 321	. . . . . . . 8 ⊢ (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
34	33	ralbidv 3188	. . . . . . 7 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0))
35		breq1 5108	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
36	35	imbi1d 344	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
37	36	ralbidv 3188	. . . . . . 7 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
38	34, 37	anbi12d 643	. . . . . 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 3584	. . . . 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 702	. . . 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 595	. . 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 787	. . . . 5 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 𝑤 ∈ ℝ*)
43		simpr 489	. . . . 5 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 0 ≤ 𝑤)
44		elxrge0 13475	. . . . 5 ⊢ (𝑤 ∈ (0[,]+∞) ↔ (𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤))
45	42, 43, 44	sylanbrc 594	. . . 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 623	. . . . . . 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 485	. . . . . 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 411	. . . . 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 485	. . . 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 5109	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑤))
52	51	notbid 321	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
53	52	ralbidv 3188	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤))
54		breq1 5108	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 < 𝑦 ↔ 𝑤 < 𝑦))
55	54	imbi1d 344	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
56	55	ralbidv 3188	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
57	53, 56	anbi12d 643	. . . . 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 3584	. . . 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 595	. . 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 780	. . . 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 13169	. . . 4 ⊢ ((𝑤 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
63	60, 61, 62	syl2anc 595	. . 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 973	. 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 3947	. . . 4 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
66	13, 65	mpan2 703	. . 3 ⊢ (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67		xrinfmss 13327	. . 3 ⊢ (𝐴 ⊆ ℝ* → ∃𝑤 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
68	66, 67	syl 18	. 2 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑤 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
69	64, 68	r19.29a 3173	1 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089   ⊆ wss 3907   class class class wbr 5105  (class class class)co 7400  0cc0 11088  +∞cpnf 11228  ℝ^*cxr 11230   < clt 11231   ≤ cle 11232  [,]cicc 13366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-po 5560  df-so 5561  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-icc 13370

This theorem is referenced by:  xrge0infssd  33018  infxrge0lb  33021  infxrge0glb  33022  infxrge0gelb  33023  omsf  34603  omssubaddlem  34606  omssubadd  34607