Theorem sup3exmid 8715

Description: If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)

Hypothesis

Ref	Expression
sup3exmid.ex	⊢ ((𝑢 ⊆ ℝ ∧ ∃𝑤 𝑤 ∈ 𝑢 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑢 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑢 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝑢 𝑦 < 𝑧)))

Assertion

Ref	Expression
sup3exmid	⊢ DECID 𝜑

Distinct variable groups:   𝑥,𝑧   𝜑,𝑢,𝑤   𝜑,𝑥,𝑦,𝑧,𝑢

Proof of Theorem sup3exmid

Dummy variables 𝑎 𝑏 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt1 7889	. . . 4 ⊢ 0 < 1
2		0re 7766	. . . . 5 ⊢ 0 ∈ ℝ
3		1re 7765	. . . . 5 ⊢ 1 ∈ ℝ
4		lttri3 7844	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 = 𝑏 ↔ (¬ 𝑎 < 𝑏 ∧ ¬ 𝑏 < 𝑎)))
5	4	adantl 275	. . . . . . 7 ⊢ ((⊤ ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑎 = 𝑏 ↔ (¬ 𝑎 < 𝑏 ∧ ¬ 𝑏 < 𝑎)))
6		elrabi 2837	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → 𝑘 ∈ {0, 1})
7		elpri 3550	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {0, 1} → (𝑘 = 0 ∨ 𝑘 = 1))
8	6, 7	syl 14	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (𝑘 = 0 ∨ 𝑘 = 1))
9		eleq1 2202	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (𝑘 ∈ ℝ ↔ 0 ∈ ℝ))
10	2, 9	mpbiri 167	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → 𝑘 ∈ ℝ)
11		eleq1 2202	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (𝑘 ∈ ℝ ↔ 1 ∈ ℝ))
12	3, 11	mpbiri 167	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → 𝑘 ∈ ℝ)
13	10, 12	jaoi 705	. . . . . . . . . . 11 ⊢ ((𝑘 = 0 ∨ 𝑘 = 1) → 𝑘 ∈ ℝ)
14	8, 13	syl 14	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → 𝑘 ∈ ℝ)
15	14	ssriv 3101	. . . . . . . . 9 ⊢ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ⊆ ℝ
16		eqid 2139	. . . . . . . . . . . 12 ⊢ 0 = 0
17	16	orci 720	. . . . . . . . . . 11 ⊢ (0 = 0 ∨ 𝜑)
18	2	elexi 2698	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
19	18	prid1 3629	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1}
20		eqeq1 2146	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 0 → (𝑗 = 0 ↔ 0 = 0))
21	20	orbi1d 780	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → ((𝑗 = 0 ∨ 𝜑) ↔ (0 = 0 ∨ 𝜑)))
22	21	elrab3 2841	. . . . . . . . . . . 12 ⊢ (0 ∈ {0, 1} → (0 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ↔ (0 = 0 ∨ 𝜑)))
23	19, 22	ax-mp 5	. . . . . . . . . . 11 ⊢ (0 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ↔ (0 = 0 ∨ 𝜑))
24	17, 23	mpbir 145	. . . . . . . . . 10 ⊢ 0 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}
25		elex2 2702	. . . . . . . . . 10 ⊢ (0 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → ∃𝑤 𝑤 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)})
26	24, 25	ax-mp 5	. . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}
27		elrabi 2837	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → 𝑦 ∈ {0, 1})
28		elpri 3550	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {0, 1} → (𝑦 = 0 ∨ 𝑦 = 1))
29		0le1 8243	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
30		breq1 3932	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 0 → (𝑦 ≤ 1 ↔ 0 ≤ 1))
31	29, 30	mpbiri 167	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → 𝑦 ≤ 1)
32	3	eqlei2 7858	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 1 → 𝑦 ≤ 1)
33	31, 32	jaoi 705	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 0 ∨ 𝑦 = 1) → 𝑦 ≤ 1)
34	28, 33	syl 14	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {0, 1} → 𝑦 ≤ 1)
35	27, 34	syl 14	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → 𝑦 ≤ 1)
36	35	rgen 2485	. . . . . . . . . 10 ⊢ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 1
37		breq2 3933	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 1))
38	37	ralbidv 2437	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 1))
39	38	rspcev 2789	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 1) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥)
40	3, 36, 39	mp2an 422	. . . . . . . . 9 ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥
41		prexg 4133	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ∈ V)
42	2, 3, 41	mp2an 422	. . . . . . . . . . 11 ⊢ {0, 1} ∈ V
43	42	rabex 4072	. . . . . . . . . 10 ⊢ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∈ V
44		sseq1 3120	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (𝑢 ⊆ ℝ ↔ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ⊆ ℝ))
45		eleq2 2203	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (𝑤 ∈ 𝑢 ↔ 𝑤 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}))
46	45	exbidv 1797	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∃𝑤 𝑤 ∈ 𝑢 ↔ ∃𝑤 𝑤 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}))
47		raleq 2626	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∀𝑦 ∈ 𝑢 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥))
48	47	rexbidv 2438	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑢 𝑦 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥))
49	44, 46, 48	3anbi123d 1290	. . . . . . . . . . 11 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → ((𝑢 ⊆ ℝ ∧ ∃𝑤 𝑤 ∈ 𝑢 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑢 𝑦 ≤ 𝑥) ↔ ({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ⊆ ℝ ∧ ∃𝑤 𝑤 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥)))
50		raleq 2626	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∀𝑦 ∈ 𝑢 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ¬ 𝑥 < 𝑦))
51		rexeq 2627	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∃𝑧 ∈ 𝑢 𝑦 < 𝑧 ↔ ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧))
52	51	imbi2d 229	. . . . . . . . . . . . . 14 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝑢 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧)))
53	52	ralbidv 2437	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝑢 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧)))
54	50, 53	anbi12d 464	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → ((∀𝑦 ∈ 𝑢 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝑢 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧))))
55	54	rexbidv 2438	. . . . . . . . . . 11 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑢 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝑢 𝑦 < 𝑧)) ↔ ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧))))
56	49, 55	imbi12d 233	. . . . . . . . . 10 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (((𝑢 ⊆ ℝ ∧ ∃𝑤 𝑤 ∈ 𝑢 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑢 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑢 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝑢 𝑦 < 𝑧))) ↔ (({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ⊆ ℝ ∧ ∃𝑤 𝑤 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧)))))
57		sup3exmid.ex	. . . . . . . . . 10 ⊢ ((𝑢 ⊆ ℝ ∧ ∃𝑤 𝑤 ∈ 𝑢 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑢 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑢 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝑢 𝑦 < 𝑧)))
58	43, 56, 57	vtocl 2740	. . . . . . . . 9 ⊢ (({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ⊆ ℝ ∧ ∃𝑤 𝑤 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧)))
59	15, 26, 40, 58	mp3an 1315	. . . . . . . 8 ⊢ ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧))
60	59	a1i 9	. . . . . . 7 ⊢ (⊤ → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧)))
61	5, 60	supclti 6885	. . . . . 6 ⊢ (⊤ → sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) ∈ ℝ)
62	61	mptru 1340	. . . . 5 ⊢ sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) ∈ ℝ
63		axltwlin 7832	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) ∈ ℝ) → (0 < 1 → (0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) ∨ sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1)))
64	2, 3, 62, 63	mp3an 1315	. . . 4 ⊢ (0 < 1 → (0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) ∨ sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1))
65	1, 64	ax-mp 5	. . 3 ⊢ (0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) ∨ sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1)
66	5, 60	suplubti 6887	. . . . . . . 8 ⊢ (⊤ → ((0 ∈ ℝ ∧ 0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < )) → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}0 < 𝑧))
67	66	mptru 1340	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < )) → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}0 < 𝑧)
68	2, 67	mpan 420	. . . . . 6 ⊢ (0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}0 < 𝑧)
69		df-rex 2422	. . . . . 6 ⊢ (∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}0 < 𝑧 ↔ ∃𝑧(𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ 0 < 𝑧))
70	68, 69	sylib 121	. . . . 5 ⊢ (0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) → ∃𝑧(𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ 0 < 𝑧))
71		eqeq1 2146	. . . . . . . . 9 ⊢ (𝑗 = 𝑧 → (𝑗 = 0 ↔ 𝑧 = 0))
72	71	orbi1d 780	. . . . . . . 8 ⊢ (𝑗 = 𝑧 → ((𝑗 = 0 ∨ 𝜑) ↔ (𝑧 = 0 ∨ 𝜑)))
73	72	elrab 2840	. . . . . . 7 ⊢ (𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ↔ (𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)))
74		simpr 109	. . . . . . . . . 10 ⊢ (((𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)) ∧ 0 < 𝑧) → 0 < 𝑧)
75	74	gt0ne0d 8274	. . . . . . . . 9 ⊢ (((𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)) ∧ 0 < 𝑧) → 𝑧 ≠ 0)
76	75	neneqd 2329	. . . . . . . 8 ⊢ (((𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)) ∧ 0 < 𝑧) → ¬ 𝑧 = 0)
77		simplr 519	. . . . . . . 8 ⊢ (((𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)) ∧ 0 < 𝑧) → (𝑧 = 0 ∨ 𝜑))
78		orel1 714	. . . . . . . 8 ⊢ (¬ 𝑧 = 0 → ((𝑧 = 0 ∨ 𝜑) → 𝜑))
79	76, 77, 78	sylc 62	. . . . . . 7 ⊢ (((𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)) ∧ 0 < 𝑧) → 𝜑)
80	73, 79	sylanb 282	. . . . . 6 ⊢ ((𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ 0 < 𝑧) → 𝜑)
81	80	exlimiv 1577	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ 0 < 𝑧) → 𝜑)
82	70, 81	syl 14	. . . 4 ⊢ (0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) → 𝜑)
83	3	ltnri 7856	. . . . . 6 ⊢ ¬ 1 < 1
84		iba 298	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∨ 𝜑) → (𝑧 ∈ {0, 1} ↔ (𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑))))
85	84	olcs 725	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ {0, 1} ↔ (𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑))))
86	85, 73	syl6rbbr 198	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ↔ 𝑧 ∈ {0, 1}))
87	86	eqrdv 2137	. . . . . . . . 9 ⊢ (𝜑 → {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} = {0, 1})
88	87	supeq1d 6874	. . . . . . . 8 ⊢ (𝜑 → sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) = sup({0, 1}, ℝ, < ))
89	3	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℝ)
90	3	elexi 2698	. . . . . . . . . . . 12 ⊢ 1 ∈ V
91	90	prid2 3630	. . . . . . . . . . 11 ⊢ 1 ∈ {0, 1}
92	91	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ {0, 1})
93		elpri 3550	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {0, 1} → (𝑧 = 0 ∨ 𝑧 = 1))
94	2, 3	lenlti 7864	. . . . . . . . . . . . . . 15 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0)
95	29, 94	mpbi 144	. . . . . . . . . . . . . 14 ⊢ ¬ 1 < 0
96		breq2 3933	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → (1 < 𝑧 ↔ 1 < 0))
97	95, 96	mtbiri 664	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → ¬ 1 < 𝑧)
98		breq2 3933	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1 → (1 < 𝑧 ↔ 1 < 1))
99	83, 98	mtbiri 664	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1 → ¬ 1 < 𝑧)
100	97, 99	jaoi 705	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∨ 𝑧 = 1) → ¬ 1 < 𝑧)
101	93, 100	syl 14	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {0, 1} → ¬ 1 < 𝑧)
102	101	adantl 275	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑧 ∈ {0, 1}) → ¬ 1 < 𝑧)
103	5, 89, 92, 102	supmaxti 6891	. . . . . . . . 9 ⊢ (⊤ → sup({0, 1}, ℝ, < ) = 1)
104	103	mptru 1340	. . . . . . . 8 ⊢ sup({0, 1}, ℝ, < ) = 1
105	88, 104	syl6eq 2188	. . . . . . 7 ⊢ (𝜑 → sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) = 1)
106	105	breq1d 3939	. . . . . 6 ⊢ (𝜑 → (sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1 ↔ 1 < 1))
107	83, 106	mtbiri 664	. . . . 5 ⊢ (𝜑 → ¬ sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1)
108	107	con2i 616	. . . 4 ⊢ (sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1 → ¬ 𝜑)
109	82, 108	orim12i 748	. . 3 ⊢ ((0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) ∨ sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1) → (𝜑 ∨ ¬ 𝜑))
110	65, 109	ax-mp 5	. 2 ⊢ (𝜑 ∨ ¬ 𝜑)
111		df-dc 820	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
112	110, 111	mpbir 145	1 ⊢ DECID 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819   ∧ w3a 962   = wceq 1331  ⊤wtru 1332  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  Vcvv 2686   ⊆ wss 3071  {cpr 3528   class class class wbr 3929  supcsup 6869  ℝcr 7619  0cc0 7620  1c1 7621   < clt 7800   ≤ cle 7801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717  ax-0lt1 7726  ax-rnegex 7729  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-iota 5088  df-riota 5730  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806

This theorem is referenced by: (None)