Step | Hyp | Ref
| Expression |
1 | | 0nn0 12231 |
. . 3
⊢ 0 ∈
ℕ0 |
2 | | simpr 484 |
. . . 4
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 = ∅) → 𝐴 = ∅) |
3 | | 0ss 4335 |
. . . 4
⊢ ∅
⊆ (0...0) |
4 | 2, 3 | eqsstrdi 3979 |
. . 3
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 = ∅) → 𝐴 ⊆ (0...0)) |
5 | | oveq2 7276 |
. . . . 5
⊢ (𝑛 = 0 → (0...𝑛) = (0...0)) |
6 | 5 | sseq2d 3957 |
. . . 4
⊢ (𝑛 = 0 → (𝐴 ⊆ (0...𝑛) ↔ 𝐴 ⊆ (0...0))) |
7 | 6 | rspcev 3560 |
. . 3
⊢ ((0
∈ ℕ0 ∧ 𝐴 ⊆ (0...0)) → ∃𝑛 ∈ ℕ0
𝐴 ⊆ (0...𝑛)) |
8 | 1, 4, 7 | sylancr 586 |
. 2
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 = ∅) → ∃𝑛 ∈ ℕ0 𝐴 ⊆ (0...𝑛)) |
9 | | elin 3907 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ↔ (𝐴 ∈ 𝒫 ℕ0 ∧
𝐴 ∈
Fin)) |
10 | 9 | simplbi 497 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫
ℕ0 ∩ Fin) → 𝐴 ∈ 𝒫
ℕ0) |
11 | 10 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ 𝒫
ℕ0) |
12 | 11 | elpwid 4549 |
. . . 4
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ⊆
ℕ0) |
13 | | nn0ssre 12220 |
. . . . . . 7
⊢
ℕ0 ⊆ ℝ |
14 | | ltso 11039 |
. . . . . . 7
⊢ < Or
ℝ |
15 | | soss 5522 |
. . . . . . 7
⊢
(ℕ0 ⊆ ℝ → ( < Or ℝ →
< Or ℕ0)) |
16 | 13, 14, 15 | mp2 9 |
. . . . . 6
⊢ < Or
ℕ0 |
17 | 16 | a1i 11 |
. . . . 5
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → < Or
ℕ0) |
18 | 9 | simprbi 496 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫
ℕ0 ∩ Fin) → 𝐴 ∈ Fin) |
19 | 18 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) |
20 | | simpr 484 |
. . . . 5
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) |
21 | | fisupcl 9189 |
. . . . 5
⊢ (( <
Or ℕ0 ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ 𝐴 ⊆ ℕ0)) →
sup(𝐴, ℕ0,
< ) ∈ 𝐴) |
22 | 17, 19, 20, 12, 21 | syl13anc 1370 |
. . . 4
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → sup(𝐴, ℕ0, < ) ∈ 𝐴) |
23 | 12, 22 | sseldd 3926 |
. . 3
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → sup(𝐴, ℕ0, < ) ∈
ℕ0) |
24 | 12 | sselda 3925 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℕ0) |
25 | | nn0uz 12602 |
. . . . . . 7
⊢
ℕ0 = (ℤ≥‘0) |
26 | 24, 25 | eleqtrdi 2850 |
. . . . . 6
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈
(ℤ≥‘0)) |
27 | 24 | nn0zd 12406 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℤ) |
28 | 12 | adantr 480 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆
ℕ0) |
29 | 22 | adantr 480 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ0, < ) ∈ 𝐴) |
30 | 28, 29 | sseldd 3926 |
. . . . . . . 8
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ0, < ) ∈
ℕ0) |
31 | 30 | nn0zd 12406 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ0, < ) ∈
ℤ) |
32 | | fisup2g 9188 |
. . . . . . . . . . . 12
⊢ (( <
Or ℕ0 ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ 𝐴 ⊆ ℕ0)) →
∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ0 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
33 | 17, 19, 20, 12, 32 | syl13anc 1370 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ0 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
34 | | ssrexv 3992 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ ℕ0
→ (∃𝑥 ∈
𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ0 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℕ0 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ0 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
35 | 12, 33, 34 | sylc 65 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ℕ0
(∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ0 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
36 | 17, 35 | supub 9179 |
. . . . . . . . 9
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → (𝑥 ∈ 𝐴 → ¬ sup(𝐴, ℕ0, < ) < 𝑥)) |
37 | 36 | imp 406 |
. . . . . . . 8
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → ¬ sup(𝐴, ℕ0, < ) < 𝑥) |
38 | 24 | nn0red 12277 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
39 | 30 | nn0red 12277 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ0, < ) ∈
ℝ) |
40 | 38, 39 | lenltd 11104 |
. . . . . . . 8
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≤ sup(𝐴, ℕ0, < ) ↔ ¬
sup(𝐴, ℕ0,
< ) < 𝑥)) |
41 | 37, 40 | mpbird 256 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℕ0, <
)) |
42 | | eluz2 12570 |
. . . . . . 7
⊢
(sup(𝐴,
ℕ0, < ) ∈ (ℤ≥‘𝑥) ↔ (𝑥 ∈ ℤ ∧ sup(𝐴, ℕ0, < ) ∈ ℤ
∧ 𝑥 ≤ sup(𝐴, ℕ0, <
))) |
43 | 27, 31, 41, 42 | syl3anbrc 1341 |
. . . . . 6
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ0, < ) ∈
(ℤ≥‘𝑥)) |
44 | | eluzfz 13233 |
. . . . . 6
⊢ ((𝑥 ∈
(ℤ≥‘0) ∧ sup(𝐴, ℕ0, < ) ∈
(ℤ≥‘𝑥)) → 𝑥 ∈ (0...sup(𝐴, ℕ0, <
))) |
45 | 26, 43, 44 | syl2anc 583 |
. . . . 5
⊢ (((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (0...sup(𝐴, ℕ0, <
))) |
46 | 45 | ex 412 |
. . . 4
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (0...sup(𝐴, ℕ0, <
)))) |
47 | 46 | ssrdv 3931 |
. . 3
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ (0...sup(𝐴, ℕ0, <
))) |
48 | | oveq2 7276 |
. . . . 5
⊢ (𝑛 = sup(𝐴, ℕ0, < ) →
(0...𝑛) = (0...sup(𝐴, ℕ0, <
))) |
49 | 48 | sseq2d 3957 |
. . . 4
⊢ (𝑛 = sup(𝐴, ℕ0, < ) → (𝐴 ⊆ (0...𝑛) ↔ 𝐴 ⊆ (0...sup(𝐴, ℕ0, <
)))) |
50 | 49 | rspcev 3560 |
. . 3
⊢
((sup(𝐴,
ℕ0, < ) ∈ ℕ0 ∧ 𝐴 ⊆ (0...sup(𝐴, ℕ0, < ))) →
∃𝑛 ∈
ℕ0 𝐴
⊆ (0...𝑛)) |
51 | 23, 47, 50 | syl2anc 583 |
. 2
⊢ ((𝐴 ∈ (𝒫
ℕ0 ∩ Fin) ∧ 𝐴 ≠ ∅) → ∃𝑛 ∈ ℕ0
𝐴 ⊆ (0...𝑛)) |
52 | 8, 51 | pm2.61dane 3033 |
1
⊢ (𝐴 ∈ (𝒫
ℕ0 ∩ Fin) → ∃𝑛 ∈ ℕ0 𝐴 ⊆ (0...𝑛)) |