zsupssdc - Intuitionistic Logic Explorer

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Theorem zsupssdc 12094

Description: An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7995.) (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 4033	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝑥 ↔ 𝑚 ≤ 𝑥))
3	2	cbvralvw 2730	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥)
4		breq2 4034	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑚 ≤ 𝑥 ↔ 𝑚 ≤ 𝑛))
5	4	ralbidv 2494	. . . . 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 2731	. . 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 2254	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
11	10	cbvexv 1930	. . . . . . 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 9615	. . . . . . 7 ⊢ (ℤ_≥‘-𝑛) ⊆ ℤ
15		rabss2 3263	. . . . . . 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 8214	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → -𝑏 = -𝑤)
18	17	eleq1d 2262	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑤 → (-𝑏 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴))
19		simp1rl 1064	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℤ)
20	19	znegcld 9444	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ∈ ℤ)
21		simp2 1000	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)
22	21	zred 9442	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
23	19	zred 9442	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℝ)
24		breq1 4033	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = -𝑤 → (𝑚 ≤ 𝑛 ↔ -𝑤 ≤ 𝑛))
25		simp1rr 1065	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
26		simp3 1001	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴)
27	24, 25, 26	rspcdva 2870	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ≤ 𝑛)
28	22, 23, 27	lenegcon1d 8548	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ≤ 𝑤)
29		eluz2 9601	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (ℤ_≥‘-𝑛) ↔ (-𝑛 ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ -𝑛 ≤ 𝑤))
30	20, 21, 28, 29	syl3anbrc 1183	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ (ℤ_≥‘-𝑛))
31	18, 30, 26	elrabd 2919	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ {𝑏 ∈ (ℤ_≥‘-𝑛) ∣ -𝑏 ∈ 𝐴})
32	31	rabssdv 3260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ {𝑏 ∈ (ℤ_≥‘-𝑛) ∣ -𝑏 ∈ 𝐴})
33	18	cbvrabv 2759	. . . . . . . . . . 11 ⊢ {𝑏 ∈ (ℤ_≥‘-𝑛) ∣ -𝑏 ∈ 𝐴} = {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}
34	32, 33	sseqtrdi 3228	. . . . . . . . . 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 3197	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} = {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
37	36	infeq1d 7073	. . . . . . . 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 529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → 𝑛 ∈ ℤ)
40	39	znegcld 9444	. . . . . . . . 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 2193	. . . . . . . 8 ⊢ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴} = {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}
43		negeq 8214	. . . . . . . . . 10 ⊢ (𝑤 = -𝑎 → -𝑤 = --𝑎)
44	43	eleq1d 2262	. . . . . . . . 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 3181	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℤ)
49	48	znegcld 9444	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑎 ∈ ℤ)
50		breq1 4033	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑎 → (𝑚 ≤ 𝑛 ↔ 𝑎 ≤ 𝑛))
51		simplrr 536	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
52	50, 51, 47	rspcdva 2870	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ 𝑛)
53	48	zred 9442	. . . . . . . . . . . 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 9442	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑛 ∈ ℝ)
56	53, 55	lenegd 8545	. . . . . . . . . . 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 9601	. . . . . . . . . 10 ⊢ (-𝑎 ∈ (ℤ_≥‘-𝑛) ↔ (-𝑛 ∈ ℤ ∧ -𝑎 ∈ ℤ ∧ -𝑛 ≤ -𝑎))
59	41, 49, 57, 58	syl3anbrc 1183	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑎 ∈ (ℤ_≥‘-𝑛))
60	48	zcnd 9443	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℂ)
61	60	negnegd 8323	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → --𝑎 = 𝑎)
62	61, 47	eqeltrd 2270	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → --𝑎 ∈ 𝐴)
63	44, 59, 62	elrabd 2919	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑎 ∈ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
64		eleq1 2256	. . . . . . . . . . 11 ⊢ (𝑥 = -𝑤 → (𝑥 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴))
65	64	dcbid 839	. . . . . . . . . 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 10094	. . . . . . . . . . . 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 9444	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → -𝑤 ∈ ℤ)
71	65, 67, 70	rspcdva 2870	. . . . . . . . 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 12091	. . . . . . 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 2270	. . . . . 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 3178	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
76	13, 75	exlimddv 1910	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
77		negeq 8214	. . . . . . 7 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → -𝑛 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
78	77	eleq1d 2262	. . . . . 6 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (-𝑛 ∈ 𝐴 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴))
79		negeq 8214	. . . . . . . 8 ⊢ (𝑤 = 𝑛 → -𝑤 = -𝑛)
80	79	eleq1d 2262	. . . . . . 7 ⊢ (𝑤 = 𝑛 → (-𝑤 ∈ 𝐴 ↔ -𝑛 ∈ 𝐴))
81	80	cbvrabv 2759	. . . . . 6 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} = {𝑛 ∈ ℤ ∣ -𝑛 ∈ 𝐴}
82	78, 81	elrab2 2920	. . . . 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 3265	. . . . . . . . 9 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ ℤ
86	85, 75	sselid 3178	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ)
87	86	zred 9442	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
88	87	renegcld 8401	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
89	41, 42, 63, 72	infssuzledc 12090	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑎)
90	38, 89	eqbrtrd 4052	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑎)
91	87, 53, 90	lenegcon2d 8549	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
92	53, 88, 91	lensymd 8143	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎)
93	92	ralrimiva 2567	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑎 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎)
94		breq2 4034	. . . . . 6 ⊢ (𝑎 = 𝑦 → (-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
95	94	notbid 668	. . . . 5 ⊢ (𝑎 = 𝑦 → (¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎 ↔ ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
96	95	cbvralv 2726	. . . 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 4034	. . . . . . 7 ⊢ (𝑧 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑧 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
99	98	rspcev 2865	. . . . . 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 2568	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑦 ∈ 𝐵 (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
103		breq1 4033	. . . . . . 7 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑥 < 𝑦 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
104	103	notbid 668	. . . . . 6 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (¬ 𝑥 < 𝑦 ↔ ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
105	104	ralbidv 2494	. . . . 5 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
106		breq2 4034	. . . . . . 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 2494	. . . . 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 2865	. . 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 1247	. 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 2616	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 835 ∧ w3a 980 = wceq 1364 ∃wex 1503 ∈ wcel 2164 ∀wral 2472 ∃wrex 2473 {crab 2476 ⊆ wss 3154 class class class wbr 4030 ‘cfv 5255 (class class class)co 5919 infcinf 7044 ℝcr 7873 < clt 8056 ≤ cle 8057 -cneg 8193 ℤcz 9320 ℤ_≥cuz 9595 ...cfz 10077

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-po 4328 df-iso 4329 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-inn 8985 df-n0 9244 df-z 9321 df-uz 9596 df-fz 10078 df-fzo 10212

This theorem is referenced by: suprzcl2dc 12095

W3C validator