zsupss - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > zsupss		Structured version Visualization version GIF version

Theorem zsupss 12606

Description: Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 10880.) (Contributed by Mario Carneiro, 21-Apr-2015.)

**Assertion**
Ref	Expression
zsupss	⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))

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

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

Step	Hyp	Ref	Expression
1		breq1 5073	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝑥 ↔ 𝑚 ≤ 𝑥))
2	1	cbvralvw 3372	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥)
3		breq2 5074	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑚 ≤ 𝑥 ↔ 𝑚 ≤ 𝑛))
4	3	ralbidv 3120	. . . . 5 ⊢ (𝑥 = 𝑛 → (∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
5	2, 4	syl5bb 282	. . . 4 ⊢ (𝑥 = 𝑛 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
6	5	cbvrexvw 3373	. . 3 ⊢ (∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑛 ∈ ℤ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
7		simp1rl 1236	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℤ)
8	7	znegcld 12357	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ∈ ℤ)
9		simp2 1135	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)
10	9	zred 12355	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
11	7	zred 12355	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℝ)
12		breq1 5073	. . . . . . . . . . 11 ⊢ (𝑚 = -𝑤 → (𝑚 ≤ 𝑛 ↔ -𝑤 ≤ 𝑛))
13		simp1rr 1237	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
14		simp3 1136	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴)
15	12, 13, 14	rspcdva 3554	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ≤ 𝑛)
16	10, 11, 15	lenegcon1d 11487	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ≤ 𝑤)
17		eluz2 12517	. . . . . . . . 9 ⊢ (𝑤 ∈ (ℤ_≥‘-𝑛) ↔ (-𝑛 ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ -𝑛 ≤ 𝑤))
18	8, 9, 16, 17	syl3anbrc 1341	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ (ℤ_≥‘-𝑛))
19	18	rabssdv 4004	. . . . . . 7 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛))
20		n0 4277	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑛 𝑛 ∈ 𝐴)
21		ssel2 3912	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℤ)
22	21	znegcld 12357	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → -𝑛 ∈ ℤ)
23	21	zcnd 12356	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℂ)
24	23	negnegd 11253	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → --𝑛 = 𝑛)
25		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ 𝐴)
26	24, 25	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → --𝑛 ∈ 𝐴)
27		negeq 11143	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = -𝑛 → -𝑤 = --𝑛)
28	27	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = -𝑛 → (-𝑤 ∈ 𝐴 ↔ --𝑛 ∈ 𝐴))
29	28	rspcev 3552	. . . . . . . . . . . . . 14 ⊢ ((-𝑛 ∈ ℤ ∧ --𝑛 ∈ 𝐴) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
30	22, 26, 29	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
31	30	ex 412	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℤ → (𝑛 ∈ 𝐴 → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴))
32	31	exlimdv 1937	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℤ → (∃𝑛 𝑛 ∈ 𝐴 → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴))
33	32	imp 406	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑛 𝑛 ∈ 𝐴) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
34	20, 33	sylan2b 593	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
35	34	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
36		rabn0 4316	. . . . . . . 8 ⊢ ({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ≠ ∅ ↔ ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
37	35, 36	sylibr 233	. . . . . . 7 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ≠ ∅)
38		infssuzcl 12601	. . . . . . 7 ⊢ (({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛) ∧ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ≠ ∅) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
39	19, 37, 38	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
40		negeq 11143	. . . . . . . . 9 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → -𝑛 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
41	40	eleq1d 2823	. . . . . . . 8 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (-𝑛 ∈ 𝐴 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴))
42		negeq 11143	. . . . . . . . . 10 ⊢ (𝑤 = 𝑛 → -𝑤 = -𝑛)
43	42	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑤 = 𝑛 → (-𝑤 ∈ 𝐴 ↔ -𝑛 ∈ 𝐴))
44	43	cbvrabv 3416	. . . . . . . 8 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} = {𝑛 ∈ ℤ ∣ -𝑛 ∈ 𝐴}
45	41, 44	elrab2 3620	. . . . . . 7 ⊢ (inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ↔ (inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ ∧ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴))
46	45	simprbi 496	. . . . . 6 ⊢ (inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴)
47	39, 46	syl 17	. . . . 5 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴)
48		simpll 763	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → 𝐴 ⊆ ℤ)
49	48	sselda 3917	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℤ)
50	49	zred 12355	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
51		ssrab2 4009	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ ℤ
52	39	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
53	51, 52	sselid 3915	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ)
54	53	znegcld 12357	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ)
55	54	zred 12355	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
56	53	zred 12355	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
57	19	adantr 480	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛))
58		negeq 11143	. . . . . . . . . . 11 ⊢ (𝑤 = -𝑦 → -𝑤 = --𝑦)
59	58	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑤 = -𝑦 → (-𝑤 ∈ 𝐴 ↔ --𝑦 ∈ 𝐴))
60	49	znegcld 12357	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -𝑦 ∈ ℤ)
61	49	zcnd 12356	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
62	61	negnegd 11253	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → --𝑦 = 𝑦)
63		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
64	62, 63	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → --𝑦 ∈ 𝐴)
65	59, 60, 64	elrabd 3619	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -𝑦 ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
66		infssuzle 12600	. . . . . . . . 9 ⊢ (({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛) ∧ -𝑦 ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑦)
67	57, 65, 66	syl2anc 583	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑦)
68	56, 50, 67	lenegcon2d 11488	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
69	50, 55, 68	lensymd 11056	. . . . . 6 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
70	69	ralrimiva 3107	. . . . 5 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
71		breq2 5074	. . . . . . . . 9 ⊢ (𝑧 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑧 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
72	71	rspcev 3552	. . . . . . . 8 ⊢ ((-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴 ∧ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
73	72	ex 412	. . . . . . 7 ⊢ (-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴 → (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
74	47, 73	syl 17	. . . . . 6 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
75	74	ralrimivw 3108	. . . . 5 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑦 ∈ 𝐵 (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
76		breq1 5073	. . . . . . . . 9 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑥 < 𝑦 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
77	76	notbid 317	. . . . . . . 8 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (¬ 𝑥 < 𝑦 ↔ ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
78	77	ralbidv 3120	. . . . . . 7 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
79		breq2 5074	. . . . . . . . 9 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑥 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
80	79	imbi1d 341	. . . . . . . 8 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
81	80	ralbidv 3120	. . . . . . 7 ⊢ (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)))
82	78, 81	anbi12d 630	. . . . . 6 ⊢ (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))))
83	82	rspcev 3552	. . . . 5 ⊢ ((-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)))
84	47, 70, 75, 83	syl12anc 833	. . . 4 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
85	84	rexlimdvaa 3213	. . 3 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) → (∃𝑛 ∈ ℤ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
86	6, 85	syl5bi 241	. 2 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))))
87	86	3impia 1115	1 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 ⊆ wss 3883 ∅c0 4253 class class class wbr 5070 ‘cfv 6418 infcinf 9130 ℝcr 10801 < clt 10940 ≤ cle 10941 -cneg 11136 ℤcz 12249 ℤ_≥cuz 12511

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512

This theorem is referenced by: suprzcl2 12607 suprzub 12608 uzsupss 12609

W3C validator