Theorem xrge0infss 32690

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 3944	. . . . . . 7 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ (0[,]+∞))
2		0xr 11228	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
3		pnfxr 11235	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
4		iccgelb 13370	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ (0[,]+∞)) → 0 ≤ 𝑦)
5	2, 3, 4	mp3an12 1453	. . . . . . . 8 ⊢ (𝑦 ∈ (0[,]+∞) → 0 ≤ 𝑦)
6		eliccxr 13403	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
7		xrlenlt 11246	. . . . . . . . 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 3126	. . . . 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 13398	. . . . . . . . . 10 ⊢ (0[,]+∞) ⊆ ℝ*
14		ssralv 4018	. . . . . . . . . 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 3947	. . . . . . . . . . . . 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 13127	. . . . . . . . . . . 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 3146	. . . . . . . . 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 13427	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
32		breq2 5114	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑦 < 𝑥 ↔ 𝑦 < 0))
33	32	notbid 318	. . . . . . . 8 ⊢ (𝑥 = 0 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0))
34	33	ralbidv 3157	. . . . . . 7 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 0))
35		breq1 5113	. . . . . . . . 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 3157	. . . . . . 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 3591	. . . . 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 13425	. . . . 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 5114	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑤))
52	51	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤))
53	52	ralbidv 3157	. . . . . 6 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤))
54		breq1 5113	. . . . . . . 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 3157	. . . . . 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 3591	. . . 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 13120	. . . 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 3958	. . . 4 ⊢ ((𝐴 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐴 ⊆ ℝ*)
66	13, 65	mpan2 691	. . 3 ⊢ (𝐴 ⊆ (0[,]+∞) → 𝐴 ⊆ ℝ*)
67		xrinfmss 13277	. . 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 3142	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 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917   class class class wbr 5110  (class class class)co 7390  0cc0 11075  +∞cpnf 11212  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216  [,]cicc 13316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-icc 13320

This theorem is referenced by:  xrge0infssd  32691  infxrge0lb  32694  infxrge0glb  32695  infxrge0gelb  32696  omsf  34294  omssubaddlem  34297  omssubadd  34298