zsupssdc - Intuitionistic Logic Explorer

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Theorem zsupssdc 11854

Description: An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7856.) (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 3970	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝑥 ↔ 𝑚 ≤ 𝑥))
3	2	cbvralvw 2684	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥)
4		breq2 3971	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑚 ≤ 𝑥 ↔ 𝑚 ≤ 𝑛))
5	4	ralbidv 2457	. . . . 5 ⊢ (𝑥 = 𝑛 → (∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
6	3, 5	syl5bb 191	. . . 4 ⊢ (𝑥 = 𝑛 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
7	6	cbvrexvw 2685	. . 3 ⊢ (∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑛 ∈ ℤ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
8	1, 7	sylib 121	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℤ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
9		zsupssdc.m	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
10		eleq1w 2218	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
11	10	cbvexv 1898	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴)
12	9, 11	sylib 121	. . . . . 6 ⊢ (𝜑 → ∃𝑎 𝑎 ∈ 𝐴)
13	12	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∃𝑎 𝑎 ∈ 𝐴)
14		uzssz 9464	. . . . . . 7 ⊢ (ℤ_≥‘-𝑛) ⊆ ℤ
15		rabss2 3211	. . . . . . 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 8073	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → -𝑏 = -𝑤)
18	17	eleq1d 2226	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑤 → (-𝑏 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴))
19		simp1rl 1047	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℤ)
20	19	znegcld 9294	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ∈ ℤ)
21		simp2 983	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)
22	21	zred 9292	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
23	19	zred 9292	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℝ)
24		breq1 3970	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = -𝑤 → (𝑚 ≤ 𝑛 ↔ -𝑤 ≤ 𝑛))
25		simp1rr 1048	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
26		simp3 984	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴)
27	24, 25, 26	rspcdva 2821	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ≤ 𝑛)
28	22, 23, 27	lenegcon1d 8407	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ≤ 𝑤)
29		eluz2 9451	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (ℤ_≥‘-𝑛) ↔ (-𝑛 ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ -𝑛 ≤ 𝑤))
30	20, 21, 28, 29	syl3anbrc 1166	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ (ℤ_≥‘-𝑛))
31	18, 30, 26	elrabd 2870	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ {𝑏 ∈ (ℤ_≥‘-𝑛) ∣ -𝑏 ∈ 𝐴})
32	31	rabssdv 3208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ {𝑏 ∈ (ℤ_≥‘-𝑛) ∣ -𝑏 ∈ 𝐴})
33	18	cbvrabv 2711	. . . . . . . . . . 11 ⊢ {𝑏 ∈ (ℤ_≥‘-𝑛) ∣ -𝑏 ∈ 𝐴} = {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}
34	32, 33	sseqtrdi 3176	. . . . . . . . . 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 3145	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} = {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
37	36	infeq1d 6959	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) = inf({𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
38	37	adantr 274	. . . . . . 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 521	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → 𝑛 ∈ ℤ)
40	39	znegcld 9294	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → -𝑛 ∈ ℤ)
41	40	adantr 274	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑛 ∈ ℤ)
42		eqid 2157	. . . . . . . 8 ⊢ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴} = {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}
43		negeq 8073	. . . . . . . . . 10 ⊢ (𝑤 = -𝑎 → -𝑤 = --𝑎)
44	43	eleq1d 2226	. . . . . . . . 9 ⊢ (𝑤 = -𝑎 → (-𝑤 ∈ 𝐴 ↔ --𝑎 ∈ 𝐴))
45		zsupssdc.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
46	45	ad2antrr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝐴 ⊆ ℤ)
47		simpr 109	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
48	46, 47	sseldd 3129	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℤ)
49	48	znegcld 9294	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑎 ∈ ℤ)
50		breq1 3970	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑎 → (𝑚 ≤ 𝑛 ↔ 𝑎 ≤ 𝑛))
51		simplrr 526	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
52	50, 51, 47	rspcdva 2821	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ 𝑛)
53	48	zred 9292	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
54	39	adantr 274	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑛 ∈ ℤ)
55	54	zred 9292	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑛 ∈ ℝ)
56	53, 55	lenegd 8404	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → (𝑎 ≤ 𝑛 ↔ -𝑛 ≤ -𝑎))
57	52, 56	mpbid 146	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑛 ≤ -𝑎)
58		eluz2 9451	. . . . . . . . . 10 ⊢ (-𝑎 ∈ (ℤ_≥‘-𝑛) ↔ (-𝑛 ∈ ℤ ∧ -𝑎 ∈ ℤ ∧ -𝑛 ≤ -𝑎))
59	41, 49, 57, 58	syl3anbrc 1166	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑎 ∈ (ℤ_≥‘-𝑛))
60	48	zcnd 9293	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℂ)
61	60	negnegd 8182	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → --𝑎 = 𝑎)
62	61, 47	eqeltrd 2234	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → --𝑎 ∈ 𝐴)
63	44, 59, 62	elrabd 2870	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -𝑎 ∈ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
64		eleq1 2220	. . . . . . . . . . 11 ⊢ (𝑥 = -𝑤 → (𝑥 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴))
65	64	dcbid 824	. . . . . . . . . 10 ⊢ (𝑥 = -𝑤 → (DECID 𝑥 ∈ 𝐴 ↔ DECID -𝑤 ∈ 𝐴))
66		zsupssdc.dc	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)
67	66	ad2antrr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → ∀𝑥 ∈ ℤ DECID 𝑥 ∈ 𝐴)
68		elfzelz 9935	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (-𝑛...-𝑎) → 𝑤 ∈ ℤ)
69	68	adantl 275	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → 𝑤 ∈ ℤ)
70	69	znegcld 9294	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → -𝑤 ∈ ℤ)
71	65, 67, 70	rspcdva 2821	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → DECID -𝑤 ∈ 𝐴)
72	71	adantlr 469	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) ∧ 𝑤 ∈ (-𝑛...-𝑎)) → DECID -𝑤 ∈ 𝐴)
73	41, 42, 63, 72	infssuzcldc 11851	. . . . . . 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 2234	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴})
75	16, 74	sseldi 3126	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
76	13, 75	exlimddv 1878	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
77		negeq 8073	. . . . . . 7 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → -𝑛 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
78	77	eleq1d 2226	. . . . . 6 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (-𝑛 ∈ 𝐴 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴))
79		negeq 8073	. . . . . . . 8 ⊢ (𝑤 = 𝑛 → -𝑤 = -𝑛)
80	79	eleq1d 2226	. . . . . . 7 ⊢ (𝑤 = 𝑛 → (-𝑤 ∈ 𝐴 ↔ -𝑛 ∈ 𝐴))
81	80	cbvrabv 2711	. . . . . 6 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} = {𝑛 ∈ ℤ ∣ -𝑛 ∈ 𝐴}
82	78, 81	elrab2 2871	. . . . 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 273	. . . 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 3213	. . . . . . . . 9 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ ℤ
86	85, 75	sseldi 3126	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ)
87	86	zred 9292	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
88	87	renegcld 8260	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
89	41, 42, 63, 72	infssuzledc 11850	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ (ℤ_≥‘-𝑛) ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑎)
90	38, 89	eqbrtrd 3989	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑎)
91	87, 53, 90	lenegcon2d 8408	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ≤ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
92	53, 88, 91	lensymd 8002	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑎 ∈ 𝐴) → ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎)
93	92	ralrimiva 2530	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑎 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎)
94		breq2 3971	. . . . . 6 ⊢ (𝑎 = 𝑦 → (-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
95	94	notbid 657	. . . . 5 ⊢ (𝑎 = 𝑦 → (¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎 ↔ ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
96	95	cbvralv 2680	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑎 ↔ ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
97	93, 96	sylib 121	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
98		breq2 3971	. . . . . . 7 ⊢ (𝑧 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑧 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
99	98	rspcev 2816	. . . . . 6 ⊢ ((-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴 ∧ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
100	99	ex 114	. . . . 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 2531	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑦 ∈ 𝐵 (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
103		breq1 3970	. . . . . . 7 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑥 < 𝑦 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
104	103	notbid 657	. . . . . 6 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (¬ 𝑥 < 𝑦 ↔ ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
105	104	ralbidv 2457	. . . . 5 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
106		breq2 3971	. . . . . . 7 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑥 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
107	106	imbi1d 230	. . . . . 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 2457	. . . . 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 465	. . . 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 2816	. . 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 1218	. 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 2579	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 103 DECID wdc 820 ∧ w3a 963 = wceq 1335 ∃wex 1472 ∈ wcel 2128 ∀wral 2435 ∃wrex 2436 {crab 2439 ⊆ wss 3102 class class class wbr 3967 ‘cfv 5173 (class class class)co 5827 infcinf 6930 ℝcr 7734 < clt 7915 ≤ cle 7916 -cneg 8052 ℤcz 9173 ℤ_≥cuz 9445 ...cfz 9919

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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-po 4259 df-iso 4260 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-isom 5182 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-sup 6931 df-inf 6932 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-inn 8840 df-n0 9097 df-z 9174 df-uz 9446 df-fz 9920 df-fzo 10052

This theorem is referenced by: suprzcl2dc 11855

W3C validator