Theorem xrsupsslem 12137

Description: Lemma for xrsupss 12139. (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 3138	. . . . . 6 ⊢ (𝐴 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦))
2		rexeq 3139	. . . . . . . 8 ⊢ (𝐴 = ∅ → (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∃𝑧 ∈ ∅ 𝑦 < 𝑧))
3	2	imbi2d 330	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
4	3	ralbidv 2986	. . . . . 6 ⊢ (𝐴 = ∅ → (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
5	1, 4	anbi12d 747	. . . . 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 3052	. . . 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 10980	. . . . . . . 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 10085	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*)
9	8	anim1i 592	. . . . . . . . 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 3010	. . . . . . . 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 11950	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
13		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
14		pnfnlt 11962	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ* → ¬ +∞ < 𝑥)
15	14	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 = +∞) → ¬ +∞ < 𝑥)
16		breq1 4656	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = +∞ → (𝑦 < 𝑥 ↔ +∞ < 𝑥))
17	16	notbid 308	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
18	17	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 = +∞) → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥))
19	15, 18	mpbird 247	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 = +∞) → ¬ 𝑦 < 𝑥)
20	19	pm2.21d 118	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 = +∞) → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
21	20	ex 450	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ* → (𝑦 = +∞ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
22	21	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 = +∞ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
23		ssel 3597	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 ∈ ℝ))
24		mnflt 11957	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ℝ → -∞ < 𝑧)
25	23, 24	syl6 35	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → -∞ < 𝑧))
26	25	ancld 576	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ -∞ < 𝑧)))
27	26	eximdv 1846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑧(𝑧 ∈ 𝐴 ∧ -∞ < 𝑧)))
28		n0 3931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
29		df-rex 2918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑧 ∈ 𝐴 -∞ < 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ -∞ < 𝑧))
30	27, 28, 29	3imtr4g 285	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
31	30	imp 445	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)
32	31	a1d 25	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
33	32	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ 𝑦 = -∞) → (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
34		breq1 4656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = -∞ → (𝑦 < 𝑥 ↔ -∞ < 𝑥))
35		breq1 4656	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = -∞ → (𝑦 < 𝑧 ↔ -∞ < 𝑧))
36	35	rexbidv 3052	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = -∞ → (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
37	34, 36	imbi12d 334	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = -∞ → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)))
38	37	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ 𝑦 = -∞) → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)))
39	33, 38	mpbird 247	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) ∧ 𝑦 = -∞) → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
40	39	ex 450	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*) → (𝑦 = -∞ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
41	40	adantr 481	. . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . 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	syl5bi 232	. . . . . . . . . . . 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 450	. . . . . . . . . . 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 2961	. . . . . . . . . 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 589	. . . . . . . . 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 3017	. . . . . . . 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 1081	. . . . . . 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 1265	. . . . 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 2992	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
52		rexnal 2995	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
53		ssel2 3598	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
54		letric 10137	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 ≤ 𝑥 ∨ 𝑥 ≤ 𝑦))
55	54	ord 392	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (¬ 𝑦 ≤ 𝑥 → 𝑥 ≤ 𝑦))
56	53, 55	sylan 488	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑦 ≤ 𝑥 → 𝑥 ≤ 𝑦))
57	56	an32s 846	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ≤ 𝑥 → 𝑥 ≤ 𝑦))
58	57	reximdva 3017	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥 → ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
59	52, 58	syl5bir 233	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
60	59	ralimdva 2962	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → (∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
61	60	imp 445	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
62	51, 61	sylan2br 493	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
63		breq2 4657	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑧))
64	63	cbvrexv 3172	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧)
65	64	ralbii 2980	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧)
66	62, 65	sylib 208	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧)
67		pnfxr 10092	. . . . . . . 8 ⊢ +∞ ∈ ℝ*
68		ssel 3597	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ))
69		rexr 10085	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
70		pnfnlt 11962	. . . . . . . . . . . . 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 2965	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)
74	73	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)
75		peano2re 10209	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
76		breq1 4656	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 ≤ 𝑧 ↔ (𝑦 + 1) ≤ 𝑧))
77	76	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (𝑦 + 1) → (∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ↔ ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧))
78	77	rspcva 3307	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 + 1) ∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧)
79	78	adantrr 753	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 + 1) ∈ ℝ ∧ (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ)) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧)
80	79	ancoms 469	. . . . . . . . . . . . . . . . . . 19 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ (𝑦 + 1) ∈ ℝ) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧)
81	75, 80	sylan2 491	. . . . . . . . . . . . . . . . . 18 ⊢ (((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧)
82		ssel2 3598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
83		ltp1 10861	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℝ → 𝑦 < (𝑦 + 1))
84	83	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑦 < (𝑦 + 1))
85	75	ancli 574	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℝ → (𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ))
86		ltletr 10129	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 < (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑧) → 𝑦 < 𝑧))
87	86	3expa 1265	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) ∧ 𝑧 ∈ ℝ) → ((𝑦 < (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑧) → 𝑦 < 𝑧))
88	85, 87	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 < (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑧) → 𝑦 < 𝑧))
89	84, 88	mpand 711	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧))
90	89	ancoms 469	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧))
91	82, 90	sylan 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧))
92	91	an32s 846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧))
93	92	reximdva 3017	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
94	93	adantll 750	. . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . 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 11953	. . . . . . . . . . . . . . . . . 18 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
100	67, 99	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ¬ +∞ < +∞
101		breq1 4656	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = +∞ → (𝑦 < +∞ ↔ +∞ < +∞))
102	100, 101	mtbiri 317	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = +∞ → ¬ 𝑦 < +∞)
103	102	pm2.21d 118	. . . . . . . . . . . . . . 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 10040	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
106		breq1 4656	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑧 ↔ 0 ≤ 𝑧))
107	106	rexbidv 3052	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 0 → (∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ↔ ∃𝑧 ∈ 𝐴 0 ≤ 𝑧))
108	107	rspcva 3307	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∃𝑧 ∈ 𝐴 0 ≤ 𝑧)
109	105, 108	mpan 706	. . . . . . . . . . . . . . . . . 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 3017	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 ∈ 𝐴 0 ≤ 𝑧 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))
113	109, 112	mpan9 486	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)
114	113, 36	syl5ibr 236	. . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . 14 ⊢ (𝑦 = -∞ → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
117	98, 104, 116	3jaoi 1391	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) → (∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
118	12, 117	sylbi 207	. . . . . . . . . . . 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 445	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → (𝑦 ∈ ℝ* → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
121	120	ralrimiv 2965	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
122	74, 121	jca 554	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
123		breq1 4656	. . . . . . . . . . . 12 ⊢ (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦))
124	123	notbid 308	. . . . . . . . . . 11 ⊢ (𝑥 = +∞ → (¬ 𝑥 < 𝑦 ↔ ¬ +∞ < 𝑦))
125	124	ralbidv 2986	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦))
126		breq2 4657	. . . . . . . . . . . 12 ⊢ (𝑥 = +∞ → (𝑦 < 𝑥 ↔ 𝑦 < +∞))
127	126	imbi1d 331	. . . . . . . . . . 11 ⊢ (𝑥 = +∞ → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
128	127	ralbidv 2986	. . . . . . . . . 10 ⊢ (𝑥 = +∞ → (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
129	125, 128	anbi12d 747	. . . . . . . . 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 3309	. . . . . . . 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 695	. . . . . . 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 487	. . . . . 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 751	. . . . 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 832	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
135		mnfxr 10096	. . . . . 6 ⊢ -∞ ∈ ℝ*
136		ral0 4076	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ ¬ -∞ < 𝑦
137		nltmnf 11963	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ* → ¬ 𝑦 < -∞)
138	137	pm2.21d 118	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))
139	138	rgen 2922	. . . . . . 7 ⊢ ∀𝑦 ∈ ℝ* (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)
140	136, 139	pm3.2i 471	. . . . . 6 ⊢ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))
141		breq1 4656	. . . . . . . . . 10 ⊢ (𝑥 = -∞ → (𝑥 < 𝑦 ↔ -∞ < 𝑦))
142	141	notbid 308	. . . . . . . . 9 ⊢ (𝑥 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ -∞ < 𝑦))
143	142	ralbidv 2986	. . . . . . . 8 ⊢ (𝑥 = -∞ → (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ ∅ ¬ -∞ < 𝑦))
144		breq2 4657	. . . . . . . . . 10 ⊢ (𝑥 = -∞ → (𝑦 < 𝑥 ↔ 𝑦 < -∞))
145	144	imbi1d 331	. . . . . . . . 9 ⊢ (𝑥 = -∞ → ((𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧) ↔ (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
146	145	ralbidv 2986	. . . . . . . 8 ⊢ (𝑥 = -∞ → (∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ* (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))
147	143, 146	anbi12d 747	. . . . . . 7 ⊢ (𝑥 = -∞ → ((∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))))
148	147	rspcev 3309	. . . . . 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 708	. . . . 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 2879	. . 3 ⊢ (𝐴 ⊆ ℝ → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
152	151	adantl 482	. 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ⊆ ℝ) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
153		ssel 3597	. . . . . 6 ⊢ (𝐴 ⊆ ℝ* → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ*))
154	153, 70	syl6 35	. . . . 5 ⊢ (𝐴 ⊆ ℝ* → (𝑦 ∈ 𝐴 → ¬ +∞ < 𝑦))
155	154	ralrimiv 2965	. . . 4 ⊢ (𝐴 ⊆ ℝ* → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)
156		breq2 4657	. . . . . . 7 ⊢ (𝑧 = +∞ → (𝑦 < 𝑧 ↔ 𝑦 < +∞))
157	156	rspcev 3309	. . . . . 6 ⊢ ((+∞ ∈ 𝐴 ∧ 𝑦 < +∞) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
158	157	ex 450	. . . . 5 ⊢ (+∞ ∈ 𝐴 → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
159	158	ralrimivw 2967	. . . 4 ⊢ (+∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
160	155, 159	anim12i 590	. . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
161	67, 160, 130	sylancr 695	. 2 ⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
162	152, 161	jaodan 826	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 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939  +∞cpnf 10071  -∞cmnf 10072  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075

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-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-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

This theorem is referenced by:  xrsupss  12139