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Theorem zsupss 12956

Description: Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 11173.) (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 5112	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝑥 ↔ 𝑚 ≤ 𝑥))
2	1	cbvralvw 3243	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥)
3		breq2 5113	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑚 ≤ 𝑥 ↔ 𝑚 ≤ 𝑛))
4	3	ralbidv 3188	. . . . 5 ⊢ (𝑥 = 𝑛 → (∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
5	2, 4	bitrid 286	. . . 4 ⊢ (𝑥 = 𝑛 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
6	5	cbvrexvw 3244	. . 3 ⊢ (∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑛 ∈ ℤ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
7		simp1rl 1257	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℤ)
8	7	znegcld 12697	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ∈ ℤ)
9		simp2 1155	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)
10	9	zred 12695	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
11	7	zred 12695	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℝ)
12		breq1 5112	. . . . . . . . . . 11 ⊢ (𝑚 = -𝑤 → (𝑚 ≤ 𝑛 ↔ -𝑤 ≤ 𝑛))
13		simp1rr 1258	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
14		simp3 1156	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴)
15	12, 13, 14	rspcdva 3582	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ≤ 𝑛)
16	10, 11, 15	lenegcon1d 11791	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ≤ 𝑤)
17		eluz2 12863	. . . . . . . . 9 ⊢ (𝑤 ∈ (ℤ_≥‘-𝑛) ↔ (-𝑛 ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ -𝑛 ≤ 𝑤))
18	8, 9, 16, 17	syl3anbrc 1362	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ (ℤ_≥‘-𝑛))
19	18	rabssdv 4028	. . . . . . 7 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛))
20		n0 4307	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑛 𝑛 ∈ 𝐴)
21		ssel2 3932	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℤ)
22	21	znegcld 12697	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → -𝑛 ∈ ℤ)
23	21	zcnd 12696	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℂ)
24	23	negnegd 11555	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → --𝑛 = 𝑛)
25		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ 𝐴)
26	24, 25	eqeltrd 2863	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → --𝑛 ∈ 𝐴)
27		negeq 11444	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = -𝑛 → -𝑤 = --𝑛)
28	27	eleq1d 2848	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = -𝑛 → (-𝑤 ∈ 𝐴 ↔ --𝑛 ∈ 𝐴))
29	28	rspcev 3581	. . . . . . . . . . . . . 14 ⊢ ((-𝑛 ∈ ℤ ∧ --𝑛 ∈ 𝐴) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
30	22, 26, 29	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
31	30	ex 417	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℤ → (𝑛 ∈ 𝐴 → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴))
32	31	exlimdv 1963	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℤ → (∃𝑛 𝑛 ∈ 𝐴 → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴))
33	32	imp 411	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑛 𝑛 ∈ 𝐴) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
34	20, 33	sylan2b 605	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
35	34	adantr 485	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
36		rabn0 4346	. . . . . . . 8 ⊢ ({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ≠ ∅ ↔ ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
37	35, 36	sylibr 237	. . . . . . 7 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ≠ ∅)
38		infssuzcl 12951	. . . . . . 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 595	. . . . . 6 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
40		negeq 11444	. . . . . . . . 9 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → -𝑛 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
41	40	eleq1d 2848	. . . . . . . 8 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (-𝑛 ∈ 𝐴 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴))
42		negeq 11444	. . . . . . . . . 10 ⊢ (𝑤 = 𝑛 → -𝑤 = -𝑛)
43	42	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑤 = 𝑛 → (-𝑤 ∈ 𝐴 ↔ -𝑛 ∈ 𝐴))
44	43	cbvrabv 3426	. . . . . . . 8 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} = {𝑛 ∈ ℤ ∣ -𝑛 ∈ 𝐴}
45	41, 44	elrab2 3654	. . . . . . 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 502	. . . . . 6 ⊢ (inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴)
47	39, 46	syl 18	. . . . 5 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴)
48		simpll 778	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → 𝐴 ⊆ ℤ)
49	48	sselda 3937	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℤ)
50	49	zred 12695	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
51		ssrab2 4034	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ ℤ
52	39	adantr 485	. . . . . . . . . 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 3935	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ)
54	53	znegcld 12697	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ)
55	54	zred 12695	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
56	53	zred 12695	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
57	19	adantr 485	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛))
58		negeq 11444	. . . . . . . . . . 11 ⊢ (𝑤 = -𝑦 → -𝑤 = --𝑦)
59	58	eleq1d 2848	. . . . . . . . . 10 ⊢ (𝑤 = -𝑦 → (-𝑤 ∈ 𝐴 ↔ --𝑦 ∈ 𝐴))
60	49	znegcld 12697	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -𝑦 ∈ ℤ)
61	49	zcnd 12696	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
62	61	negnegd 11555	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → --𝑦 = 𝑦)
63		simpr 489	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
64	62, 63	eqeltrd 2863	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → --𝑦 ∈ 𝐴)
65	59, 60, 64	elrabd 3652	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -𝑦 ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
66		infssuzle 12950	. . . . . . . . 9 ⊢ (({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛) ∧ -𝑦 ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑦)
67	57, 65, 66	syl2anc 595	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑦)
68	56, 50, 67	lenegcon2d 11792	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
69	50, 55, 68	lensymd 11356	. . . . . 6 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
70	69	ralrimiva 3157	. . . . 5 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
71		breq2 5113	. . . . . . . . 9 ⊢ (𝑧 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑧 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
72	71	rspcev 3581	. . . . . . . 8 ⊢ ((-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴 ∧ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
73	72	ex 417	. . . . . . 7 ⊢ (-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴 → (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
74	47, 73	syl 18	. . . . . 6 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → (𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))
75	74	ralrimivw 3161	. . . . 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 5112	. . . . . . . . 9 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑥 < 𝑦 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
77	76	notbid 321	. . . . . . . 8 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (¬ 𝑥 < 𝑦 ↔ ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
78	77	ralbidv 3188	. . . . . . 7 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
79		breq2 5113	. . . . . . . . 9 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑥 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
80	79	imbi1d 344	. . . . . . . 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 3188	. . . . . . 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 643	. . . . . 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 3581	. . . . 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 849	. . . 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 3167	. . 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	biimtrid 245	. 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 1135	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 400 ∧ w3a 1103 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 {crab 3416 ⊆ wss 3905 ∅c0 4286 class class class wbr 5109 ‘cfv 6536 infcinf 9397 ℝcr 11094 < clt 11238 ≤ cle 11239 -cneg 11437 ℤcz 12586 ℤ_≥cuz 12857

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9398 df-inf 9399 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-n0 12500 df-z 12587 df-uz 12858

This theorem is referenced by: suprzcl2 12957 suprzub 12958 uzsupss 12959

W3C validator