| Step | Hyp | Ref
| Expression |
| 1 | | 1nn 12277 |
. . 3
⊢ 1 ∈
ℕ |
| 2 | | simpr 484 |
. . . 4
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 = ∅)
→ 𝐴 =
∅) |
| 3 | | 0ss 4400 |
. . . 4
⊢ ∅
⊆ (1...1) |
| 4 | 2, 3 | eqsstrdi 4028 |
. . 3
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 = ∅)
→ 𝐴 ⊆
(1...1)) |
| 5 | | oveq2 7439 |
. . . . 5
⊢ (𝑛 = 1 → (1...𝑛) = (1...1)) |
| 6 | 5 | sseq2d 4016 |
. . . 4
⊢ (𝑛 = 1 → (𝐴 ⊆ (1...𝑛) ↔ 𝐴 ⊆ (1...1))) |
| 7 | 6 | rspcev 3622 |
. . 3
⊢ ((1
∈ ℕ ∧ 𝐴
⊆ (1...1)) → ∃𝑛 ∈ ℕ 𝐴 ⊆ (1...𝑛)) |
| 8 | 1, 4, 7 | sylancr 587 |
. 2
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 = ∅)
→ ∃𝑛 ∈
ℕ 𝐴 ⊆
(1...𝑛)) |
| 9 | | elin 3967 |
. . . . . . 7
⊢ (𝐴 ∈ (𝒫 ℕ ∩
Fin) ↔ (𝐴 ∈
𝒫 ℕ ∧ 𝐴
∈ Fin)) |
| 10 | 9 | simplbi 497 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ℕ ∩
Fin) → 𝐴 ∈
𝒫 ℕ) |
| 11 | 10 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ 𝐴 ∈ 𝒫
ℕ) |
| 12 | 11 | elpwid 4609 |
. . . 4
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ 𝐴 ⊆
ℕ) |
| 13 | | nnssre 12270 |
. . . . . . 7
⊢ ℕ
⊆ ℝ |
| 14 | | ltso 11341 |
. . . . . . 7
⊢ < Or
ℝ |
| 15 | | soss 5612 |
. . . . . . 7
⊢ (ℕ
⊆ ℝ → ( < Or ℝ → < Or
ℕ)) |
| 16 | 13, 14, 15 | mp2 9 |
. . . . . 6
⊢ < Or
ℕ |
| 17 | 16 | a1i 11 |
. . . . 5
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ < Or ℕ) |
| 18 | 9 | simprbi 496 |
. . . . . 6
⊢ (𝐴 ∈ (𝒫 ℕ ∩
Fin) → 𝐴 ∈
Fin) |
| 19 | 18 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ 𝐴 ∈
Fin) |
| 20 | | simpr 484 |
. . . . 5
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ 𝐴 ≠
∅) |
| 21 | | fisupcl 9509 |
. . . . 5
⊢ (( <
Or ℕ ∧ (𝐴 ∈
Fin ∧ 𝐴 ≠ ∅
∧ 𝐴 ⊆ ℕ))
→ sup(𝐴, ℕ, <
) ∈ 𝐴) |
| 22 | 17, 19, 20, 12, 21 | syl13anc 1374 |
. . . 4
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ sup(𝐴, ℕ, <
) ∈ 𝐴) |
| 23 | 12, 22 | sseldd 3984 |
. . 3
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ sup(𝐴, ℕ, <
) ∈ ℕ) |
| 24 | 12 | sselda 3983 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℕ) |
| 25 | | nnuz 12921 |
. . . . . . 7
⊢ ℕ =
(ℤ≥‘1) |
| 26 | 24, 25 | eleqtrdi 2851 |
. . . . . 6
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → 𝑥 ∈
(ℤ≥‘1)) |
| 27 | 24 | nnzd 12640 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℤ) |
| 28 | 12 | adantr 480 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℕ) |
| 29 | 22 | adantr 480 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ, < ) ∈ 𝐴) |
| 30 | 28, 29 | sseldd 3984 |
. . . . . . . 8
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ, < ) ∈
ℕ) |
| 31 | 30 | nnzd 12640 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ, < ) ∈
ℤ) |
| 32 | | fisup2g 9508 |
. . . . . . . . . . . 12
⊢ (( <
Or ℕ ∧ (𝐴 ∈
Fin ∧ 𝐴 ≠ ∅
∧ 𝐴 ⊆ ℕ))
→ ∃𝑥 ∈
𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
| 33 | 17, 19, 20, 12, 32 | syl13anc 1374 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ ∃𝑥 ∈
𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
| 34 | | ssrexv 4053 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ ℕ →
(∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℕ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
| 35 | 12, 33, 34 | sylc 65 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ ∃𝑥 ∈
ℕ (∀𝑦 ∈
𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
| 36 | 17, 35 | supub 9499 |
. . . . . . . . 9
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ (𝑥 ∈ 𝐴 → ¬ sup(𝐴, ℕ, < ) < 𝑥)) |
| 37 | 36 | imp 406 |
. . . . . . . 8
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → ¬ sup(𝐴, ℕ, < ) < 𝑥) |
| 38 | 24 | nnred 12281 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
| 39 | 30 | nnred 12281 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ, < ) ∈
ℝ) |
| 40 | 38, 39 | lenltd 11407 |
. . . . . . . 8
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → (𝑥 ≤ sup(𝐴, ℕ, < ) ↔ ¬ sup(𝐴, ℕ, < ) < 𝑥)) |
| 41 | 37, 40 | mpbird 257 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℕ, < )) |
| 42 | | eluz2 12884 |
. . . . . . 7
⊢
(sup(𝐴, ℕ,
< ) ∈ (ℤ≥‘𝑥) ↔ (𝑥 ∈ ℤ ∧ sup(𝐴, ℕ, < ) ∈ ℤ ∧ 𝑥 ≤ sup(𝐴, ℕ, < ))) |
| 43 | 27, 31, 41, 42 | syl3anbrc 1344 |
. . . . . 6
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℕ, < ) ∈
(ℤ≥‘𝑥)) |
| 44 | | eluzfz 13559 |
. . . . . 6
⊢ ((𝑥 ∈
(ℤ≥‘1) ∧ sup(𝐴, ℕ, < ) ∈
(ℤ≥‘𝑥)) → 𝑥 ∈ (1...sup(𝐴, ℕ, < ))) |
| 45 | 26, 43, 44 | syl2anc 584 |
. . . . 5
⊢ (((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (1...sup(𝐴, ℕ, < ))) |
| 46 | 45 | ex 412 |
. . . 4
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ (𝑥 ∈ 𝐴 → 𝑥 ∈ (1...sup(𝐴, ℕ, < )))) |
| 47 | 46 | ssrdv 3989 |
. . 3
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ 𝐴 ⊆
(1...sup(𝐴, ℕ, <
))) |
| 48 | | oveq2 7439 |
. . . . 5
⊢ (𝑛 = sup(𝐴, ℕ, < ) → (1...𝑛) = (1...sup(𝐴, ℕ, < ))) |
| 49 | 48 | sseq2d 4016 |
. . . 4
⊢ (𝑛 = sup(𝐴, ℕ, < ) → (𝐴 ⊆ (1...𝑛) ↔ 𝐴 ⊆ (1...sup(𝐴, ℕ, < )))) |
| 50 | 49 | rspcev 3622 |
. . 3
⊢
((sup(𝐴, ℕ,
< ) ∈ ℕ ∧ 𝐴 ⊆ (1...sup(𝐴, ℕ, < ))) → ∃𝑛 ∈ ℕ 𝐴 ⊆ (1...𝑛)) |
| 51 | 23, 47, 50 | syl2anc 584 |
. 2
⊢ ((𝐴 ∈ (𝒫 ℕ ∩
Fin) ∧ 𝐴 ≠ ∅)
→ ∃𝑛 ∈
ℕ 𝐴 ⊆
(1...𝑛)) |
| 52 | 8, 51 | pm2.61dane 3029 |
1
⊢ (𝐴 ∈ (𝒫 ℕ ∩
Fin) → ∃𝑛 ∈
ℕ 𝐴 ⊆
(1...𝑛)) |