zsupssdc - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > zsupssdc		GIF version

Theorem zsupssdc 10497

Description: An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8152.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)

**Hypotheses**
Ref	Expression
zsupssdc.a	⊢ (𝜑 → 𝐴 ⊆ ℤ)
zsupssdc.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
zsupssdc.dc	⊢ (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)
zsupssdc.ub	⊢ (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

**Assertion**
Ref	Expression
zsupssdc	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))

Distinct variable groups: 𝑥,𝐴,𝑦,𝑧 𝑥,𝐵 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑧) 𝐵(𝑦,𝑧)

Proof of Theorem zsupssdc Dummy variables 𝑎 𝑚 𝑛 𝑤 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zsupssdc.ub	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
2		breq1 4091	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝑥 ↔ 𝑚 ≤ 𝑥))
3	2	cbvralvw 2771	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥)
4		breq2 4092	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑚 ≤ 𝑥 ↔ 𝑚 ≤ 𝑛))
5	4	ralbidv 2532	. . . . 5 ⊢ (𝑥 = 𝑛 → (∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
6	3, 5	bitrid 192	. . . 4 ⊢ (𝑥 = 𝑛 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
7	6	cbvrexvw 2772	. . 3 ⊢ (∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑛 ∈ ℤ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
8	1, 7	sylib 122	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℤ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
9		zsupssdc.m	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
10		eleq1w 2292	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
11	10	cbvexv 1967	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴)
12	9, 11	sylib 122	. . . . . 6 ⊢ (𝜑 → ∃𝑎 𝑎 ∈ 𝐴)
13	12	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∃𝑎 𝑎 ∈ 𝐴)
14		uzssz 9775	. . . . . . 7 ⊢ (ℤ_≥‘-𝑛) ⊆ ℤ
15		rabss2 3310	. . . . . . 7 ⊢ ((ℤ_≥‘-𝑛) ⊆ ℤ → {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴} ⊆ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
16	14, 15	ax-mp 5	. . . . . 6 ⊢ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴} ⊆ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}
17		negeq 8371	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → -𝑏 = -𝑤)
18	17	eleq1d 2300	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑤 → (-𝑏 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴))
19		simp1rl 1088	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℤ)
20	19	znegcld 9603	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ∈ ℤ)
21		simp2 1024	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)
22	21	zred 9601	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
23	19	zred 9601	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℝ)
24		breq1 4091	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = -𝑤 → (𝑚 ≤ 𝑛 ↔ -𝑤 ≤ 𝑛))
25		simp1rr 1089	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
26		simp3 1025	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴)
27	24, 25, 26	rspcdva 2915	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ≤ 𝑛)
28	22, 23, 27	lenegcon1d 8706	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ≤ 𝑤)
29		eluz2 9760	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (ℤ_≥‘-𝑛) ↔ (-𝑛 ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ -𝑛 ≤ 𝑤))
30	20, 21, 28, 29	syl3anbrc 1207	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ (ℤ_≥‘-𝑛))
31	18, 30, 26	elrabd 2964	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ {𝑏 ∈ (ℤ_≥‘-𝑛) ∣ -𝑏 ∈ 𝐴})
32	31	rabssdv 3307	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ {𝑏 ∈ (ℤ_≥‘-𝑛) ∣ -𝑏 ∈ 𝐴})
33	18	cbvrabv 2801	. . . . . . . . . . 11 ⊢ {𝑏 ∈ (ℤ_≥‘-𝑛) ∣ -𝑏 ∈ 𝐴} = {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}
34	32, 33	sseqtrdi 3275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
35	16	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴} ⊆ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
36	34, 35	eqssd 3244	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} = {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
37	36	infeq1d 7210	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) = inf({𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
38	37	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) = inf({𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
39		simprl 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → 𝑛 ∈ ℤ)
40	39	znegcld 9603	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → -𝑛 ∈ ℤ)
41	40	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑛 ∈ ℤ)
42		eqid 2231	. . . . . . . 8 ⊢ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴} = {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}
43		negeq 8371	. . . . . . . . . 10 ⊢ (𝑤 = -𝑎 → -𝑤 = --𝑎)
44	43	eleq1d 2300	. . . . . . . . 9 ⊢ (𝑤 = -𝑎 → (-𝑤 ∈ 𝐴 ↔ --𝑎 ∈ 𝐴))
45		zsupssdc.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
46	45	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝐴 ⊆ ℤ)
47		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
48	46, 47	sseldd 3228	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℤ)
49	48	znegcld 9603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑎 ∈ ℤ)
50		breq1 4091	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑎 → (𝑚 ≤ 𝑛 ↔ 𝑎 ≤ 𝑛))
51		simplrr 538	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
52	50, 51, 47	rspcdva 2915	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ 𝑛)
53	48	zred 9601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
54	39	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑛 ∈ ℤ)
55	54	zred 9601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑛 ∈ ℝ)
56	53, 55	lenegd 8703	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → (𝑎 ≤ 𝑛 ↔ -𝑛 ≤ -𝑎))
57	52, 56	mpbid 147	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑛 ≤ -𝑎)
58		eluz2 9760	. . . . . . . . . 10 ⊢ (-𝑎 ∈ (ℤ_≥‘-𝑛) ↔ (-𝑛 ∈ ℤ ∧ -𝑎 ∈ ℤ ∧ -𝑛 ≤ -𝑎))
59	41, 49, 57, 58	syl3anbrc 1207	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑎 ∈ (ℤ_≥‘-𝑛))
60	48	zcnd 9602	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℂ)
61	60	negnegd 8480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → --𝑎 = 𝑎)
62	61, 47	eqeltrd 2308	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → --𝑎 ∈ 𝐴)
63	44, 59, 62	elrabd 2964	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑎 ∈ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
64		eleq1 2294	. . . . . . . . . . 11 ⊢ (𝑥 = -𝑤 → (𝑥 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴))
65	64	dcbid 845	. . . . . . . . . 10 ⊢ (𝑥 = -𝑤 → (DECID 𝑥 ∈ 𝐴 ↔ DECID -𝑤 ∈ 𝐴))
66		zsupssdc.dc	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)
67	66	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)
68		elfzelz 10259	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (-𝑛...-𝑎) → 𝑤 ∈ ℤ)
69	68	adantl 277	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → 𝑤 ∈ ℤ)
70	69	znegcld 9603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → -𝑤 ∈ ℤ)
71	65, 67, 70	rspcdva 2915	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → DECID -𝑤 ∈ 𝐴)
72	71	adantlr 477	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → DECID -𝑤 ∈ 𝐴)
73	41, 42, 63, 72	infssuzcldc 10494	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
74	38, 73	eqeltrd 2308	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
75	16, 74	sselid 3225	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
76	13, 75	exlimddv 1947	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
77		negeq 8371	. . . . . . 7 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → -𝑛 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
78	77	eleq1d 2300	. . . . . 6 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (-𝑛 ∈ 𝐴 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴))
79		negeq 8371	. . . . . . . 8 ⊢ (𝑤 = 𝑛 → -𝑤 = -𝑛)
80	79	eleq1d 2300	. . . . . . 7 ⊢ (𝑤 = 𝑛 → (-𝑤 ∈ 𝐴 ↔ -𝑛 ∈ 𝐴))
81	80	cbvrabv 2801	. . . . . 6 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} = {𝑛 ∈ ℤ ∣ -𝑛 ∈ 𝐴}
82	78, 81	elrab2 2965	. . . . 5 ⊢ (inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ↔ (inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ ∧ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴))
83	82	simprbi 275	. . . 4 ⊢ (inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴)
84	76, 83	syl 14	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴)
85		ssrab2 3312	. . . . . . . . 9 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ ℤ
86	85, 75	sselid 3225	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ)
87	86	zred 9601	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
88	87	renegcld 8558	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
89	41, 42, 63, 72	infssuzledc 10493	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑎)
90	38, 89	eqbrtrd 4110	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑎)
91	87, 53, 90	lenegcon2d 8707	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
92	53, 88, 91	lensymd 8300	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎)
93	92	ralrimiva 2605	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑎 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎)
94		breq2 4092	. . . . . 6 ⊢ (𝑎 = 𝑦 → (-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
95	94	notbid 673	. . . . 5 ⊢ (𝑎 = 𝑦 → (¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎 ↔ ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
96	95	cbvralv 2767	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎 ↔ ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
97	93, 96	sylib 122	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
98		breq2 4092	. . . . . . 7 ⊢ (𝑧 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑧 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
99	98	rspcev 2910	. . . . . 6 ⊢ ((-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴 ∧ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
100	99	ex 115	. . . . 5 ⊢ (-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴 → (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
101	84, 100	syl 14	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
102	101	ralrimivw 2606	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑦 ∈ 𝐵 (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
103		breq1 4091	. . . . . . 7 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑥 < 𝑦 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
104	103	notbid 673	. . . . . 6 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (¬ 𝑥 < 𝑦 ↔ ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
105	104	ralbidv 2532	. . . . 5 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
106		breq2 4092	. . . . . . 7 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑥 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
107	106	imbi1d 231	. . . . . 6 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
108	107	ralbidv 2532	. . . . 5 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ 𝐵 (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
109	105, 108	anbi12d 473	. . . 4 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
110	109	rspcev 2910	. . 3 ⊢ ((-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
111	84, 97, 102, 110	syl12anc 1271	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
112	8, 111	rexlimddv 2655	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 DECID wdc 841 ∧ w3a 1004 = wceq 1397 ∃wex 1540 ∈ wcel 2202 ∀wral 2510 ∃wrex 2511 {crab 2514 ⊆ wss 3200 class class class wbr 4088 ‘cfv 5326 (class class class)co 6017 infcinf 7181 ℝcr 8030 < clt 8213 ≤ cle 8214 -cneg 8350 ℤcz 9478 ℤ_≥cuz 9754 ...cfz 10242

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-addcom 8131 ax-addass 8133 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-0id 8139 ax-rnegex 8140 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-po 4393 df-iso 4394 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-inn 9143 df-n0 9402 df-z 9479 df-uz 9755 df-fz 10243 df-fzo 10377

This theorem is referenced by: suprzcl2dc 10498

W3C validator