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Theorem zsupss 12878

Description: Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 11107.) (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 5075	. . . . . 6 ⊢ (𝑦 = 𝑚 → (𝑦 ≤ 𝑥 ↔ 𝑚 ≤ 𝑥))
2	1	cbvralvw 3217	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥)
3		breq2 5076	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑚 ≤ 𝑥 ↔ 𝑚 ≤ 𝑛))
4	3	ralbidv 3162	. . . . 5 ⊢ (𝑥 = 𝑛 → (∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
5	2, 4	bitrid 284	. . . 4 ⊢ (𝑥 = 𝑛 → (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛))
6	5	cbvrexvw 3218	. . 3 ⊢ (∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑛 ∈ ℤ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
7		simp1rl 1245	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℤ)
8	7	znegcld 12626	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ∈ ℤ)
9		simp2 1143	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℤ)
10	9	zred 12624	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
11	7	zred 12624	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑛 ∈ ℝ)
12		breq1 5075	. . . . . . . . . . 11 ⊢ (𝑚 = -𝑤 → (𝑚 ≤ 𝑛 ↔ -𝑤 ≤ 𝑛))
13		simp1rr 1246	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)
14		simp3 1144	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ∈ 𝐴)
15	12, 13, 14	rspcdva 3561	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑤 ≤ 𝑛)
16	10, 11, 15	lenegcon1d 11723	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → -𝑛 ≤ 𝑤)
17		eluz2 12785	. . . . . . . . 9 ⊢ (𝑤 ∈ (ℤ_≥‘-𝑛) ↔ (-𝑛 ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ -𝑛 ≤ 𝑤))
18	8, 9, 16, 17	syl3anbrc 1350	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑤 ∈ ℤ ∧ -𝑤 ∈ 𝐴) → 𝑤 ∈ (ℤ_≥‘-𝑛))
19	18	rabssdv 4005	. . . . . . 7 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛))
20		n0 4281	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑛 𝑛 ∈ 𝐴)
21		ssel2 3910	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℤ)
22	21	znegcld 12626	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → -𝑛 ∈ ℤ)
23	21	zcnd 12625	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℂ)
24	23	negnegd 11487	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → --𝑛 = 𝑛)
25		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ 𝐴)
26	24, 25	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → --𝑛 ∈ 𝐴)
27		negeq 11376	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = -𝑛 → -𝑤 = --𝑛)
28	27	eleq1d 2824	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = -𝑛 → (-𝑤 ∈ 𝐴 ↔ --𝑛 ∈ 𝐴))
29	28	rspcev 3560	. . . . . . . . . . . . . 14 ⊢ ((-𝑛 ∈ ℤ ∧ --𝑛 ∈ 𝐴) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
30	22, 26, 29	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℤ ∧ 𝑛 ∈ 𝐴) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
31	30	ex 413	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℤ → (𝑛 ∈ 𝐴 → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴))
32	31	exlimdv 1940	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℤ → (∃𝑛 𝑛 ∈ 𝐴 → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴))
33	32	imp 407	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑛 𝑛 ∈ 𝐴) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
34	20, 33	sylan2b 600	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
35	34	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
36		rabn0 4317	. . . . . . . 8 ⊢ ({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ≠ ∅ ↔ ∃𝑤 ∈ ℤ -𝑤 ∈ 𝐴)
37	35, 36	sylibr 235	. . . . . . 7 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ≠ ∅)
38		infssuzcl 12873	. . . . . . 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 590	. . . . . 6 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
40		negeq 11376	. . . . . . . . 9 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → -𝑛 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
41	40	eleq1d 2824	. . . . . . . 8 ⊢ (𝑛 = inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (-𝑛 ∈ 𝐴 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴))
42		negeq 11376	. . . . . . . . . 10 ⊢ (𝑤 = 𝑛 → -𝑤 = -𝑛)
43	42	eleq1d 2824	. . . . . . . . 9 ⊢ (𝑤 = 𝑛 → (-𝑤 ∈ 𝐴 ↔ -𝑛 ∈ 𝐴))
44	43	cbvrabv 3401	. . . . . . . 8 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} = {𝑛 ∈ ℤ ∣ -𝑛 ∈ 𝐴}
45	41, 44	elrab2 3632	. . . . . . 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 498	. . . . . 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 772	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → 𝐴 ⊆ ℤ)
49	48	sselda 3915	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℤ)
50	49	zred 12624	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
51		ssrab2 4011	. . . . . . . . . 10 ⊢ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ ℤ
52	39	adantr 481	. . . . . . . . . 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 3913	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ)
54	53	znegcld 12626	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℤ)
55	54	zred 12624	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
56	53	zred 12624	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ ℝ)
57	19	adantr 481	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛))
58		negeq 11376	. . . . . . . . . . 11 ⊢ (𝑤 = -𝑦 → -𝑤 = --𝑦)
59	58	eleq1d 2824	. . . . . . . . . 10 ⊢ (𝑤 = -𝑦 → (-𝑤 ∈ 𝐴 ↔ --𝑦 ∈ 𝐴))
60	49	znegcld 12626	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -𝑦 ∈ ℤ)
61	49	zcnd 12625	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
62	61	negnegd 11487	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → --𝑦 = 𝑦)
63		simpr 485	. . . . . . . . . . 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 3631	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → -𝑦 ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴})
66		infssuzle 12872	. . . . . . . . 9 ⊢ (({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴} ⊆ (ℤ_≥‘-𝑛) ∧ -𝑦 ∈ {𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑦)
67	57, 65, 66	syl2anc 590	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ≤ -𝑦)
68	56, 50, 67	lenegcon2d 11724	. . . . . . 7 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ))
69	50, 55, 68	lensymd 11288	. . . . . 6 ⊢ ((((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) ∧ 𝑦 ∈ 𝐴) → ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
70	69	ralrimiva 3131	. . . . 5 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅) ∧ (𝑛 ∈ ℤ ∧ ∀𝑚 ∈ 𝐴 𝑚 ≤ 𝑛)) → ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦)
71		breq2 5076	. . . . . . . . 9 ⊢ (𝑧 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑧 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
72	71	rspcev 3560	. . . . . . . 8 ⊢ ((-inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) ∈ 𝐴 ∧ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)
73	72	ex 413	. . . . . . 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 3135	. . . . 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 5075	. . . . . . . . 9 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑥 < 𝑦 ↔ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
77	76	notbid 319	. . . . . . . 8 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (¬ 𝑥 < 𝑦 ↔ ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
78	77	ralbidv 3162	. . . . . . 7 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) < 𝑦))
79		breq2 5076	. . . . . . . . 9 ⊢ (𝑥 = -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < ) → (𝑦 < 𝑥 ↔ 𝑦 < -inf({𝑤 ∈ ℤ ∣ -𝑤 ∈ 𝐴}, ℝ, < )))
80	79	imbi1d 342	. . . . . . . 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 3162	. . . . . . 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 638	. . . . . 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 3560	. . . . 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 842	. . . 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 3141	. . 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 243	. 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 1123	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 396 ∧ w3a 1092 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 {crab 3391 ⊆ wss 3883 ∅c0 4261 class class class wbr 5072 ‘cfv 6485 infcinf 9344 ℝcr 11028 < clt 11170 ≤ cle 11171 -cneg 11369 ℤcz 12515 ℤ_≥cuz 12779

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780

This theorem is referenced by: suprzcl2 12879 suprzub 12880 uzsupss 12881

W3C validator