Step |
Hyp |
Ref |
Expression |
1 |
|
1nn |
⊢ 1 ∈ ℕ |
2 |
|
simpr |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 = ∅ ) → 𝐴 = ∅ ) |
3 |
|
0ss |
⊢ ∅ ⊆ ( 1 ... 1 ) |
4 |
2 3
|
eqsstrdi |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 = ∅ ) → 𝐴 ⊆ ( 1 ... 1 ) ) |
5 |
|
oveq2 |
⊢ ( 𝑛 = 1 → ( 1 ... 𝑛 ) = ( 1 ... 1 ) ) |
6 |
5
|
sseq2d |
⊢ ( 𝑛 = 1 → ( 𝐴 ⊆ ( 1 ... 𝑛 ) ↔ 𝐴 ⊆ ( 1 ... 1 ) ) ) |
7 |
6
|
rspcev |
⊢ ( ( 1 ∈ ℕ ∧ 𝐴 ⊆ ( 1 ... 1 ) ) → ∃ 𝑛 ∈ ℕ 𝐴 ⊆ ( 1 ... 𝑛 ) ) |
8 |
1 4 7
|
sylancr |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 = ∅ ) → ∃ 𝑛 ∈ ℕ 𝐴 ⊆ ( 1 ... 𝑛 ) ) |
9 |
|
elin |
⊢ ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ↔ ( 𝐴 ∈ 𝒫 ℕ ∧ 𝐴 ∈ Fin ) ) |
10 |
9
|
simplbi |
⊢ ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) → 𝐴 ∈ 𝒫 ℕ ) |
11 |
10
|
adantr |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → 𝐴 ∈ 𝒫 ℕ ) |
12 |
11
|
elpwid |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → 𝐴 ⊆ ℕ ) |
13 |
|
nnssre |
⊢ ℕ ⊆ ℝ |
14 |
|
ltso |
⊢ < Or ℝ |
15 |
|
soss |
⊢ ( ℕ ⊆ ℝ → ( < Or ℝ → < Or ℕ ) ) |
16 |
13 14 15
|
mp2 |
⊢ < Or ℕ |
17 |
16
|
a1i |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → < Or ℕ ) |
18 |
9
|
simprbi |
⊢ ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) → 𝐴 ∈ Fin ) |
19 |
18
|
adantr |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → 𝐴 ∈ Fin ) |
20 |
|
simpr |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → 𝐴 ≠ ∅ ) |
21 |
|
fisupcl |
⊢ ( ( < Or ℕ ∧ ( 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ 𝐴 ⊆ ℕ ) ) → sup ( 𝐴 , ℕ , < ) ∈ 𝐴 ) |
22 |
17 19 20 12 21
|
syl13anc |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → sup ( 𝐴 , ℕ , < ) ∈ 𝐴 ) |
23 |
12 22
|
sseldd |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → sup ( 𝐴 , ℕ , < ) ∈ ℕ ) |
24 |
12
|
sselda |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℕ ) |
25 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
26 |
24 25
|
eleqtrdi |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ( ℤ≥ ‘ 1 ) ) |
27 |
24
|
nnzd |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℤ ) |
28 |
12
|
adantr |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐴 ⊆ ℕ ) |
29 |
22
|
adantr |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → sup ( 𝐴 , ℕ , < ) ∈ 𝐴 ) |
30 |
28 29
|
sseldd |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → sup ( 𝐴 , ℕ , < ) ∈ ℕ ) |
31 |
30
|
nnzd |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → sup ( 𝐴 , ℕ , < ) ∈ ℤ ) |
32 |
|
fisup2g |
⊢ ( ( < 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 |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) ) |
34 |
|
ssrexv |
⊢ ( 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 |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ ℕ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) ) |
36 |
17 35
|
supub |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → ( 𝑥 ∈ 𝐴 → ¬ sup ( 𝐴 , ℕ , < ) < 𝑥 ) ) |
37 |
36
|
imp |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → ¬ sup ( 𝐴 , ℕ , < ) < 𝑥 ) |
38 |
24
|
nnred |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℝ ) |
39 |
30
|
nnred |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → sup ( 𝐴 , ℕ , < ) ∈ ℝ ) |
40 |
38 39
|
lenltd |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ≤ sup ( 𝐴 , ℕ , < ) ↔ ¬ sup ( 𝐴 , ℕ , < ) < 𝑥 ) ) |
41 |
37 40
|
mpbird |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ≤ sup ( 𝐴 , ℕ , < ) ) |
42 |
|
eluz2 |
⊢ ( sup ( 𝐴 , ℕ , < ) ∈ ( ℤ≥ ‘ 𝑥 ) ↔ ( 𝑥 ∈ ℤ ∧ sup ( 𝐴 , ℕ , < ) ∈ ℤ ∧ 𝑥 ≤ sup ( 𝐴 , ℕ , < ) ) ) |
43 |
27 31 41 42
|
syl3anbrc |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → sup ( 𝐴 , ℕ , < ) ∈ ( ℤ≥ ‘ 𝑥 ) ) |
44 |
|
eluzfz |
⊢ ( ( 𝑥 ∈ ( ℤ≥ ‘ 1 ) ∧ sup ( 𝐴 , ℕ , < ) ∈ ( ℤ≥ ‘ 𝑥 ) ) → 𝑥 ∈ ( 1 ... sup ( 𝐴 , ℕ , < ) ) ) |
45 |
26 43 44
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ( 1 ... sup ( 𝐴 , ℕ , < ) ) ) |
46 |
45
|
ex |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 1 ... sup ( 𝐴 , ℕ , < ) ) ) ) |
47 |
46
|
ssrdv |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → 𝐴 ⊆ ( 1 ... sup ( 𝐴 , ℕ , < ) ) ) |
48 |
|
oveq2 |
⊢ ( 𝑛 = sup ( 𝐴 , ℕ , < ) → ( 1 ... 𝑛 ) = ( 1 ... sup ( 𝐴 , ℕ , < ) ) ) |
49 |
48
|
sseq2d |
⊢ ( 𝑛 = sup ( 𝐴 , ℕ , < ) → ( 𝐴 ⊆ ( 1 ... 𝑛 ) ↔ 𝐴 ⊆ ( 1 ... sup ( 𝐴 , ℕ , < ) ) ) ) |
50 |
49
|
rspcev |
⊢ ( ( sup ( 𝐴 , ℕ , < ) ∈ ℕ ∧ 𝐴 ⊆ ( 1 ... sup ( 𝐴 , ℕ , < ) ) ) → ∃ 𝑛 ∈ ℕ 𝐴 ⊆ ( 1 ... 𝑛 ) ) |
51 |
23 47 50
|
syl2anc |
⊢ ( ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) ∧ 𝐴 ≠ ∅ ) → ∃ 𝑛 ∈ ℕ 𝐴 ⊆ ( 1 ... 𝑛 ) ) |
52 |
8 51
|
pm2.61dane |
⊢ ( 𝐴 ∈ ( 𝒫 ℕ ∩ Fin ) → ∃ 𝑛 ∈ ℕ 𝐴 ⊆ ( 1 ... 𝑛 ) ) |