Theorem sup3exmid 9037

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 8206	. . . 4 ⊢ 0 < 1
2		0re 8079	. . . . 5 ⊢ 0 ∈ ℝ
3		1re 8078	. . . . 5 ⊢ 1 ∈ ℝ
4		lttri3 8159	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 = 𝑏 ↔ (¬ 𝑎 < 𝑏 ∧ ¬ 𝑏 < 𝑎)))
5	4	adantl 277	. . . . . . 7 ⊢ ((⊤ ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑎 = 𝑏 ↔ (¬ 𝑎 < 𝑏 ∧ ¬ 𝑏 < 𝑎)))
6		elrabi 2927	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → 𝑘 ∈ {0, 1})
7		elpri 3657	. . . . . . . . . . . 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 2269	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (𝑘 ∈ ℝ ↔ 0 ∈ ℝ))
10	2, 9	mpbiri 168	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → 𝑘 ∈ ℝ)
11		eleq1 2269	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (𝑘 ∈ ℝ ↔ 1 ∈ ℝ))
12	3, 11	mpbiri 168	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → 𝑘 ∈ ℝ)
13	10, 12	jaoi 718	. . . . . . . . . . 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 3198	. . . . . . . . 9 ⊢ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ⊆ ℝ
16		eqid 2206	. . . . . . . . . . . 12 ⊢ 0 = 0
17	16	orci 733	. . . . . . . . . . 11 ⊢ (0 = 0 ∨ 𝜑)
18	2	elexi 2785	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
19	18	prid1 3740	. . . . . . . . . . . 12 ⊢ 0 ∈ {0, 1}
20		eqeq1 2213	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 0 → (𝑗 = 0 ↔ 0 = 0))
21	20	orbi1d 793	. . . . . . . . . . . . 13 ⊢ (𝑗 = 0 → ((𝑗 = 0 ∨ 𝜑) ↔ (0 = 0 ∨ 𝜑)))
22	21	elrab3 2931	. . . . . . . . . . . 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 146	. . . . . . . . . 10 ⊢ 0 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}
25		elex2 2789	. . . . . . . . . 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 2927	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → 𝑦 ∈ {0, 1})
28		elpri 3657	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {0, 1} → (𝑦 = 0 ∨ 𝑦 = 1))
29		0le1 8561	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
30		breq1 4050	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 0 → (𝑦 ≤ 1 ↔ 0 ≤ 1))
31	29, 30	mpbiri 168	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → 𝑦 ≤ 1)
32	3	eqlei2 8174	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 1 → 𝑦 ≤ 1)
33	31, 32	jaoi 718	. . . . . . . . . . . . 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 2560	. . . . . . . . . 10 ⊢ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 1
37		breq2 4051	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 1))
38	37	ralbidv 2507	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 1))
39	38	rspcev 2878	. . . . . . . . . 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 426	. . . . . . . . 9 ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥
41		prexg 4259	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ∈ V)
42	2, 3, 41	mp2an 426	. . . . . . . . . . 11 ⊢ {0, 1} ∈ V
43	42	rabex 4192	. . . . . . . . . 10 ⊢ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∈ V
44		sseq1 3217	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (𝑢 ⊆ ℝ ↔ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ⊆ ℝ))
45		eleq2 2270	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (𝑤 ∈ 𝑢 ↔ 𝑤 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}))
46	45	exbidv 1849	. . . . . . . . . . . 12 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∃𝑤 𝑤 ∈ 𝑢 ↔ ∃𝑤 𝑤 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}))
47		raleq 2703	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∀𝑦 ∈ 𝑢 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 ≤ 𝑥))
48	47	rexbidv 2508	. . . . . . . . . . . 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 1325	. . . . . . . . . . 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 2703	. . . . . . . . . . . . 13 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∀𝑦 ∈ 𝑢 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ¬ 𝑥 < 𝑦))
51		rexeq 2704	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} → (∃𝑧 ∈ 𝑢 𝑦 < 𝑧 ↔ ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}𝑦 < 𝑧))
52	51	imbi2d 230	. . . . . . . . . . . . . 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 2507	. . . . . . . . . . . . 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 473	. . . . . . . . . . . 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 2508	. . . . . . . . . . 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 234	. . . . . . . . . 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 2828	. . . . . . . . 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 1350	. . . . . . . 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 7107	. . . . . 6 ⊢ (⊤ → sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) ∈ ℝ)
62	61	mptru 1382	. . . . 5 ⊢ sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) ∈ ℝ
63		axltwlin 8147	. . . . 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 1350	. . . 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 7109	. . . . . . . 8 ⊢ (⊤ → ((0 ∈ ℝ ∧ 0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < )) → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}0 < 𝑧))
67	66	mptru 1382	. . . . . . 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 424	. . . . . 6 ⊢ (0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) → ∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}0 < 𝑧)
69		df-rex 2491	. . . . . 6 ⊢ (∃𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}0 < 𝑧 ↔ ∃𝑧(𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ 0 < 𝑧))
70	68, 69	sylib 122	. . . . 5 ⊢ (0 < sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) → ∃𝑧(𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ 0 < 𝑧))
71		eqeq1 2213	. . . . . . . . 9 ⊢ (𝑗 = 𝑧 → (𝑗 = 0 ↔ 𝑧 = 0))
72	71	orbi1d 793	. . . . . . . 8 ⊢ (𝑗 = 𝑧 → ((𝑗 = 0 ∨ 𝜑) ↔ (𝑧 = 0 ∨ 𝜑)))
73	72	elrab 2930	. . . . . . 7 ⊢ (𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ↔ (𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)))
74		simpr 110	. . . . . . . . . 10 ⊢ (((𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)) ∧ 0 < 𝑧) → 0 < 𝑧)
75	74	gt0ne0d 8592	. . . . . . . . 9 ⊢ (((𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)) ∧ 0 < 𝑧) → 𝑧 ≠ 0)
76	75	neneqd 2398	. . . . . . . 8 ⊢ (((𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)) ∧ 0 < 𝑧) → ¬ 𝑧 = 0)
77		simplr 528	. . . . . . . 8 ⊢ (((𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑)) ∧ 0 < 𝑧) → (𝑧 = 0 ∨ 𝜑))
78		orel1 727	. . . . . . . 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 284	. . . . . 6 ⊢ ((𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ∧ 0 < 𝑧) → 𝜑)
81	80	exlimiv 1622	. . . . 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 8172	. . . . . 6 ⊢ ¬ 1 < 1
84		iba 300	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∨ 𝜑) → (𝑧 ∈ {0, 1} ↔ (𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑))))
85	84	olcs 738	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ {0, 1} ↔ (𝑧 ∈ {0, 1} ∧ (𝑧 = 0 ∨ 𝜑))))
86	73, 85	bitr4id 199	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} ↔ 𝑧 ∈ {0, 1}))
87	86	eqrdv 2204	. . . . . . . . 9 ⊢ (𝜑 → {𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)} = {0, 1})
88	87	supeq1d 7096	. . . . . . . 8 ⊢ (𝜑 → sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) = sup({0, 1}, ℝ, < ))
89	3	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℝ)
90	3	elexi 2785	. . . . . . . . . . . 12 ⊢ 1 ∈ V
91	90	prid2 3741	. . . . . . . . . . 11 ⊢ 1 ∈ {0, 1}
92	91	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ {0, 1})
93		elpri 3657	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {0, 1} → (𝑧 = 0 ∨ 𝑧 = 1))
94	2, 3	lenlti 8180	. . . . . . . . . . . . . . 15 ⊢ (0 ≤ 1 ↔ ¬ 1 < 0)
95	29, 94	mpbi 145	. . . . . . . . . . . . . 14 ⊢ ¬ 1 < 0
96		breq2 4051	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → (1 < 𝑧 ↔ 1 < 0))
97	95, 96	mtbiri 677	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → ¬ 1 < 𝑧)
98		breq2 4051	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1 → (1 < 𝑧 ↔ 1 < 1))
99	83, 98	mtbiri 677	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1 → ¬ 1 < 𝑧)
100	97, 99	jaoi 718	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∨ 𝑧 = 1) → ¬ 1 < 𝑧)
101	93, 100	syl 14	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {0, 1} → ¬ 1 < 𝑧)
102	101	adantl 277	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑧 ∈ {0, 1}) → ¬ 1 < 𝑧)
103	5, 89, 92, 102	supmaxti 7113	. . . . . . . . 9 ⊢ (⊤ → sup({0, 1}, ℝ, < ) = 1)
104	103	mptru 1382	. . . . . . . 8 ⊢ sup({0, 1}, ℝ, < ) = 1
105	88, 104	eqtrdi 2255	. . . . . . 7 ⊢ (𝜑 → sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) = 1)
106	105	breq1d 4057	. . . . . 6 ⊢ (𝜑 → (sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1 ↔ 1 < 1))
107	83, 106	mtbiri 677	. . . . 5 ⊢ (𝜑 → ¬ sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1)
108	107	con2i 628	. . . 4 ⊢ (sup({𝑗 ∈ {0, 1} ∣ (𝑗 = 0 ∨ 𝜑)}, ℝ, < ) < 1 → ¬ 𝜑)
109	82, 108	orim12i 761	. . 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 837	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
112	110, 111	mpbir 146	1 ⊢ DECID 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710  DECID wdc 836   ∧ w3a 981   = wceq 1373  ⊤wtru 1374  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  {crab 2489  Vcvv 2773   ⊆ wss 3167  {cpr 3635   class class class wbr 4047  supcsup 7091  ℝcr 7931  0cc0 7932  1c1 7933   < clt 8114   ≤ cle 8115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029  ax-0lt1 8038  ax-rnegex 8041  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-xp 4685  df-cnv 4687  df-iota 5237  df-riota 5906  df-sup 7093  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120

This theorem is referenced by: (None)