Theorem xrge0infss 29525

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 3598	. . . . . . 7 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (0[,]+∞))
2		0xr 10086	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
3		pnfxr 10092	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
4		iccgelb 12230	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
5	2, 3, 4	mp3an12 1414	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6		iccssxr 12256	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ*
7	6	sseli 3599	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
8		xrlenlt 10103	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
9	2, 7, 8	sylancr 695	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
10	5, 9	mpbid 222	. . . . . . 7 ⊢ (𝑦 ∈ (0[,]+∞) → ¬ 𝑦 < 0)
11	1, 10	syl 17	. . . . . 6 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 < 0)
12	11	ralrimiva 2966	. . . . 5 ⊢ (𝐴 ⊆ (0[,]+∞) → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0)
13	12	ad3antrrr 766	. . . 4 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 𝑤 ≤ 0) → ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0)
14		ssralv 3666	. . . . . . . . . 10 ⊢ ((0[,]+∞) ⊆ ℝ* → (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
15	6, 14	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) → ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))
16		simplll 798	. . . . . . . . . . . . 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 792	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ (0[,]+∞))
19	6, 18	sseldi 3601	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑦 ∈ ℝ*)
20		simpllr 799	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 ≤ 0)
21		simpr 477	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 0 < 𝑦)
22	16, 17, 19, 20, 21	xrlelttrd 11991	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0) ∧ 𝑦 ∈ (0[,]+∞)) ∧ 0 < 𝑦) → 𝑤 < 𝑦)
23	22	ex 450	. . . . . . . . . . 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 2962	. . . . . . . . 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 750	. . . . . . 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 445	. . . . . 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 752	. . . . 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 846	. . . 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 12283	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
32		breq2 4657	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑦 < 𝑥 ↔ 𝑦 < 0))
33	32	notbid 308	. . . . . . . 8 ⊢ (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
34	33	ralbidv 2986	. . . . . . 7 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0))
35		breq1 4656	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
36	35	imbi1d 331	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
37	36	ralbidv 2986	. . . . . . 7 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(0 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
38	34, 37	anbi12d 747	. . . . . 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 3309	. . . . 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 706	. . . 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	13, 30, 40	syl2anc 693	. . 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 799	. . . . 5 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 𝑤 ∈ ℝ*)
43		simpr 477	. . . . 5 ⊢ ((((𝐴 ⊆ (0[,]+∞) ∧ 𝑤 ∈ ℝ*) ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀𝑦 ∈ ℝ* (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) ∧ 0 ≤ 𝑤) → 0 ≤ 𝑤)
44		elxrge0 12281	. . . . 5 ⊢ (𝑤 ∈ (0[,]+∞) ↔ (𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤))
45	42, 43, 44	sylanbrc 698	. . . 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 589	. . . . . . 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 481	. . . . . 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 445	. . . . 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 481	. . . 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 4657	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑤))
52	51	notbid 308	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
53	52	ralbidv 2986	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤))
54		breq1 4656	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 < 𝑦 ↔ 𝑤 < 𝑦))
55	54	imbi1d 331	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
56	55	ralbidv 2986	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ (0[,]+∞)(𝑤 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
57	53, 56	anbi12d 747	. . . . 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 3309	. . . 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 693	. . 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 792	. . . 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 11984	. . . 4 ⊢ ((𝑤 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝑤 ≤ 0 ∨ 0 ≤ 𝑤))
63	60, 61, 62	syl2anc 693	. . 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 827	. 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 3611	. . . 4 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
66	6, 65	mpan2 707	. . 3 ⊢ (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67		xrinfmss 12140	. . 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 3078	1 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   class class class wbr 4653  (class class class)co 6650  0cc0 9936  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  [,]cicc 12178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-icc 12182

This theorem is referenced by:  xrge0infssd  29526  infxrge0lb  29529  infxrge0glb  29530  infxrge0gelb  29531  omsf  30358  omssubaddlem  30361  omssubadd  30362