Theorem xrsupsslem 13209

Description: Lemma for xrsupss 13211. (Contributed by NM, 25-Oct-2005.)

Assertion

Ref	Expression
xrsupsslem	⊢ ((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem xrsupsslem

Step	Hyp	Ref	Expression
1		raleq 3286	. . . . . 6 ⊢ (𝐴 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦))
2		rexeq 3285	. . . . . . . 8 ⊢ (𝐴 = ∅ → (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∃𝑧 ∈ ∅ 𝑦 < 𝑧))
3	2	imbi2d 340	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
4	3	ralbidv 3152	. . . . . 6 ⊢ (𝐴 = ∅ → (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
5	1, 4	anbi12d 632	. . . . 5 ⊢ (𝐴 = ∅ → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))))
6	5	rexbidv 3153	. . . 4 ⊢ (𝐴 = ∅ → (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ ∃𝑥 ∈ ℝ* (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))))
7		sup3 12082	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
8		rexr 11161	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
9	8	anim1i 615	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑥 ∈ ℝ* ∧ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
10	9	reximi2 3062	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
11	7, 10	syl 17	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
12		elxr 13018	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
13		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
14		pnfnlt 13030	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ* → ¬ +∞ < 𝑥)
15	14	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 = +∞) → ¬ +∞ < 𝑥)
16		breq1 5095	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = +∞ → (𝑦 < 𝑥 ↔ +∞ < 𝑥))
17	16	notbid 318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
18	17	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 = +∞) → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
19	15, 18	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 = +∞) → ¬ 𝑦 < 𝑥)
20	19	pm2.21d 121	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 = +∞) → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
21	20	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ* → (𝑦 = +∞ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
22	21	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 = +∞ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
23		ssel 3929	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 ∈ ℝ))
24		mnflt 13025	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℝ → -∞ < 𝑧)
25	23, 24	syl6 35	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → -∞ < 𝑧))
26	25	ancld 550	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ -∞ < 𝑧)))
27	26	eximdv 1917	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑧(𝑧 ∈ 𝐴 ∧ -∞ < 𝑧)))
28		n0 4304	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
29		df-rex 3054	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑧 ∈ 𝐴 -∞ < 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ -∞ < 𝑧))
30	27, 28, 29	3imtr4g 296	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
31	30	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)
32	31	a1d 25	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
33	32	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ 𝑦 = -∞) → (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
34		breq1 5095	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = -∞ → (𝑦 < 𝑥 ↔ -∞ < 𝑥))
35		breq1 5095	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = -∞ → (𝑦 < 𝑧 ↔ -∞ < 𝑧))
36	35	rexbidv 3153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = -∞ → (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
37	34, 36	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = -∞ → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)))
38	37	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ 𝑦 = -∞) → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)))
39	33, 38	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
40	39	ex 412	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) → (𝑦 = -∞ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
41	40	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 = -∞ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
42	13, 22, 41	3jaod 1431	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
43	12, 42	biimtrid 242	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 ∈ ℝ* → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
44	43	ex 412	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) → ((𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → (𝑦 ∈ ℝ* → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
45	44	ralimdv2 3138	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
46	45	anim2d 612	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
47	46	reximdva 3142	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
48	47	3adant3 1132	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
49	11, 48	mpd 15	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
50	49	3expa 1118	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
51		ralnex 3055	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
52		rexnal 3081	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
53		ssel2 3930	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
54		letric 11216	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 ≤ 𝑥 ∨ 𝑥 ≤ 𝑦))
55	54	ord 864	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (¬ 𝑦 ≤ 𝑥 → 𝑥 ≤ 𝑦))
56	53, 55	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑦 ≤ 𝑥 → 𝑥 ≤ 𝑦))
57	56	an32s 652	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ≤ 𝑥 → 𝑥 ≤ 𝑦))
58	57	reximdva 3142	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥 → ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
59	52, 58	biimtrrid 243	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
60	59	ralimdva 3141	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
61	60	imp 406	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
62	51, 61	sylan2br 595	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
63		breq2 5096	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑧))
64	63	cbvrexvw 3208	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧)
65	64	ralbii 3075	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧)
66	62, 65	sylib 218	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧)
67		pnfxr 11169	. . . . . . . 8 ⊢ +∞ ∈ ℝ*
68		ssel 3929	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ))
69		rexr 11161	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
70		pnfnlt 13030	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → ¬ +∞ < 𝑦)
71	69, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → ¬ +∞ < 𝑦)
72	68, 71	syl6 35	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → ¬ +∞ < 𝑦))
73	72	ralrimiv 3120	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)
74	73	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)
75		peano2re 11289	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
76		breq1 5095	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 ≤ 𝑧 ↔ (𝑦 + 1) ≤ 𝑧))
77	76	rexbidv 3153	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑦 + 1) → (∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ↔ ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧))
78	77	rspcva 3575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 + 1) ∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧)
79	78	adantrr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1) ∈ ℝ ∧ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ)) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧)
80	79	ancoms 458	. . . . . . . . . . . . . . . . . . 19 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ (𝑦 + 1) ∈ ℝ) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧)
81	75, 80	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧)
82		ssel2 3930	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
83		ltp1 11964	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℝ → 𝑦 < (𝑦 + 1))
84	83	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑦 < (𝑦 + 1))
85	75	ancli 548	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ))
86		ltletr 11208	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 < (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑧) → 𝑦 < 𝑧))
87	86	3expa 1118	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) ∧ 𝑧 ∈ ℝ) → ((𝑦 < (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑧) → 𝑦 < 𝑧))
88	85, 87	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 < (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑧) → 𝑦 < 𝑧))
89	84, 88	mpand 695	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧))
90	89	ancoms 458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧))
91	82, 90	sylan 580	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧))
92	91	an32s 652	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧))
93	92	reximdva 3142	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
94	93	adantll 714	. . . . . . . . . . . . . . . . . 18 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
95	81, 94	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
96	95	exp31 419	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 ∈ ℝ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
97	96	a1dd 50	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → (𝑦 ∈ ℝ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
98	97	com4r 94	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
99		xrltnr 13021	. . . . . . . . . . . . . . . . . 18 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
100	67, 99	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ¬ +∞ < +∞
101		breq1 5095	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = +∞ → (𝑦 < +∞ ↔ +∞ < +∞))
102	100, 101	mtbiri 327	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = +∞ → ¬ 𝑦 < +∞)
103	102	pm2.21d 121	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = +∞ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
104	103	2a1d 26	. . . . . . . . . . . . . 14 ⊢ (𝑦 = +∞ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
105		0re 11117	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
106		breq1 5095	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑧 ↔ 0 ≤ 𝑧))
107	106	rexbidv 3153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 0 → (∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ↔ ∃𝑧 ∈ 𝐴 0 ≤ 𝑧))
108	107	rspcva 3575	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∃𝑧 ∈ 𝐴 0 ≤ 𝑧)
109	105, 108	mpan 690	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → ∃𝑧 ∈ 𝐴 0 ≤ 𝑧)
110	82, 24	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → -∞ < 𝑧)
111	110	a1d 25	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → (0 ≤ 𝑧 → -∞ < 𝑧))
112	111	reximdva 3142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 ∈ 𝐴 0 ≤ 𝑧 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
113	109, 112	mpan9 506	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)
114	113, 36	imbitrrid 246	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = -∞ → ((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
115	114	a1dd 50	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = -∞ → ((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
116	115	expd 415	. . . . . . . . . . . . . 14 ⊢ (𝑦 = -∞ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
117	98, 104, 116	3jaoi 1430	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
118	12, 117	sylbi 217	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
119	118	com13 88	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝑦 ∈ ℝ* → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
120	119	imp 406	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → (𝑦 ∈ ℝ* → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
121	120	ralrimiv 3120	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
122	74, 121	jca 511	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
123		breq1 5095	. . . . . . . . . . . 12 ⊢ (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
124	123	notbid 318	. . . . . . . . . . 11 ⊢ (𝑥 = +∞ → (¬ 𝑥 < 𝑦 ↔ ¬ +∞ < 𝑦))
125	124	ralbidv 3152	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦))
126		breq2 5096	. . . . . . . . . . . 12 ⊢ (𝑥 = +∞ → (𝑦 < 𝑥 ↔ 𝑦 < +∞))
127	126	imbi1d 341	. . . . . . . . . . 11 ⊢ (𝑥 = +∞ → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
128	127	ralbidv 3152	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
129	125, 128	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = +∞ → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
130	129	rspcev 3577	. . . . . . . 8 ⊢ ((+∞ ∈ ℝ* ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
131	67, 122, 130	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
132	66, 131	syldan 591	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
133	132	adantlr 715	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
134	50, 133	pm2.61dan 812	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
135		mnfxr 11172	. . . . . 6 ⊢ -∞ ∈ ℝ*
136		ral0 4464	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ ¬ -∞ < 𝑦
137		nltmnf 13031	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ* → ¬ 𝑦 < -∞)
138	137	pm2.21d 121	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))
139	138	rgen 3046	. . . . . . 7 ⊢ ∀𝑦 ∈ ℝ* (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)
140	136, 139	pm3.2i 470	. . . . . 6 ⊢ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))
141		breq1 5095	. . . . . . . . . 10 ⊢ (𝑥 = -∞ → (𝑥 < 𝑦 ↔ -∞ < 𝑦))
142	141	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ -∞ < 𝑦))
143	142	ralbidv 3152	. . . . . . . 8 ⊢ (𝑥 = -∞ → (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ ∅ ¬ -∞ < 𝑦))
144		breq2 5096	. . . . . . . . . 10 ⊢ (𝑥 = -∞ → (𝑦 < 𝑥 ↔ 𝑦 < -∞))
145	144	imbi1d 341	. . . . . . . . 9 ⊢ (𝑥 = -∞ → ((𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧) ↔ (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
146	145	ralbidv 3152	. . . . . . . 8 ⊢ (𝑥 = -∞ → (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ* (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
147	143, 146	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = -∞ → ((∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))))
148	147	rspcev 3577	. . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
149	135, 140, 148	mp2an 692	. . . . 5 ⊢ ∃𝑥 ∈ ℝ* (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))
150	149	a1i 11	. . . 4 ⊢ (𝐴 ⊆ ℝ → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
151	6, 134, 150	pm2.61ne 3010	. . 3 ⊢ (𝐴 ⊆ ℝ → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
152	151	adantl 481	. 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ⊆ ℝ) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
153		ssel 3929	. . . . . 6 ⊢ (𝐴 ⊆ ℝ* → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ*))
154	153, 70	syl6 35	. . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝑦 ∈ 𝐴 → ¬ +∞ < 𝑦))
155	154	ralrimiv 3120	. . . 4 ⊢ (𝐴 ⊆ ℝ* → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)
156		breq2 5096	. . . . . . 7 ⊢ (𝑧 = +∞ → (𝑦 < 𝑧 ↔ 𝑦 < +∞))
157	156	rspcev 3577	. . . . . 6 ⊢ ((+∞ ∈ 𝐴 ∧ 𝑦 < +∞) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
158	157	ex 412	. . . . 5 ⊢ (+∞ ∈ 𝐴 → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
159	158	ralrimivw 3125	. . . 4 ⊢ (+∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
160	155, 159	anim12i 613	. . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
161	67, 160, 130	sylancr 587	. 2 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
162	152, 161	jaodan 959	1 ⊢ ((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ⊆ wss 3903  ∅c0 4284   class class class wbr 5092  (class class class)co 7349  ℝcr 11008  0cc0 11009  1c1 11010   + caddc 11012  +∞cpnf 11146  -∞cmnf 11147  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350

This theorem is referenced by:  xrsupss  13211