Step |
Hyp |
Ref |
Expression |
1 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
2 |
1
|
a1i |
⊢ ( 𝑆 = ∅ → 0 ∈ ℕ0 ) |
3 |
|
breq1 |
⊢ ( 𝑠 = 0 → ( 𝑠 < 𝑥 ↔ 0 < 𝑥 ) ) |
4 |
3
|
imbi1d |
⊢ ( 𝑠 = 0 → ( ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ↔ ( 0 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
5 |
4
|
ralbidv |
⊢ ( 𝑠 = 0 → ( ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ↔ ∀ 𝑥 ∈ ℕ0 ( 0 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝑆 = ∅ ∧ 𝑠 = 0 ) → ( ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ↔ ∀ 𝑥 ∈ ℕ0 ( 0 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
7 |
|
nnel |
⊢ ( ¬ 𝑥 ∉ 𝑆 ↔ 𝑥 ∈ 𝑆 ) |
8 |
|
n0i |
⊢ ( 𝑥 ∈ 𝑆 → ¬ 𝑆 = ∅ ) |
9 |
7 8
|
sylbi |
⊢ ( ¬ 𝑥 ∉ 𝑆 → ¬ 𝑆 = ∅ ) |
10 |
9
|
con4i |
⊢ ( 𝑆 = ∅ → 𝑥 ∉ 𝑆 ) |
11 |
10
|
a1d |
⊢ ( 𝑆 = ∅ → ( 0 < 𝑥 → 𝑥 ∉ 𝑆 ) ) |
12 |
11
|
ralrimivw |
⊢ ( 𝑆 = ∅ → ∀ 𝑥 ∈ ℕ0 ( 0 < 𝑥 → 𝑥 ∉ 𝑆 ) ) |
13 |
2 6 12
|
rspcedvd |
⊢ ( 𝑆 = ∅ → ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) |
14 |
13
|
2a1d |
⊢ ( 𝑆 = ∅ → ( 𝑆 ⊆ ℕ0 → ( 𝑆 ∈ Fin → ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) ) |
15 |
|
ltso |
⊢ < Or ℝ |
16 |
|
id |
⊢ ( 𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℕ0 ) |
17 |
|
nn0ssre |
⊢ ℕ0 ⊆ ℝ |
18 |
16 17
|
sstrdi |
⊢ ( 𝑆 ⊆ ℕ0 → 𝑆 ⊆ ℝ ) |
19 |
18
|
3anim3i |
⊢ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) → ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ ) ) |
20 |
|
fisup2g |
⊢ ( ( < Or ℝ ∧ ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℝ ) ) → ∃ 𝑠 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑠 → ∃ 𝑧 ∈ 𝑆 𝑦 < 𝑧 ) ) ) |
21 |
15 19 20
|
sylancr |
⊢ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) → ∃ 𝑠 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑠 → ∃ 𝑧 ∈ 𝑆 𝑦 < 𝑧 ) ) ) |
22 |
|
simp3 |
⊢ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) → 𝑆 ⊆ ℕ0 ) |
23 |
|
breq2 |
⊢ ( 𝑦 = 𝑥 → ( 𝑠 < 𝑦 ↔ 𝑠 < 𝑥 ) ) |
24 |
23
|
notbid |
⊢ ( 𝑦 = 𝑥 → ( ¬ 𝑠 < 𝑦 ↔ ¬ 𝑠 < 𝑥 ) ) |
25 |
24
|
rspcva |
⊢ ( ( 𝑥 ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ) → ¬ 𝑠 < 𝑥 ) |
26 |
25
|
2a1d |
⊢ ( ( 𝑥 ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ) → ( 𝑥 ∈ ℕ0 → ( ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) → ¬ 𝑠 < 𝑥 ) ) ) |
27 |
26
|
expcom |
⊢ ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → ( 𝑥 ∈ 𝑆 → ( 𝑥 ∈ ℕ0 → ( ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) → ¬ 𝑠 < 𝑥 ) ) ) ) |
28 |
27
|
com24 |
⊢ ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → ( ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) → ( 𝑥 ∈ ℕ0 → ( 𝑥 ∈ 𝑆 → ¬ 𝑠 < 𝑥 ) ) ) ) |
29 |
28
|
imp31 |
⊢ ( ( ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) ) ∧ 𝑥 ∈ ℕ0 ) → ( 𝑥 ∈ 𝑆 → ¬ 𝑠 < 𝑥 ) ) |
30 |
7 29
|
syl5bi |
⊢ ( ( ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) ) ∧ 𝑥 ∈ ℕ0 ) → ( ¬ 𝑥 ∉ 𝑆 → ¬ 𝑠 < 𝑥 ) ) |
31 |
30
|
con4d |
⊢ ( ( ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) ) ∧ 𝑥 ∈ ℕ0 ) → ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) |
32 |
31
|
ralrimiva |
⊢ ( ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) ) → ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) |
33 |
32
|
ex |
⊢ ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 → ( ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) → ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
34 |
33
|
adantr |
⊢ ( ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑠 → ∃ 𝑧 ∈ 𝑆 𝑦 < 𝑧 ) ) → ( ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) → ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
35 |
34
|
com12 |
⊢ ( ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) ∧ 𝑠 ∈ 𝑆 ) → ( ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑠 → ∃ 𝑧 ∈ 𝑆 𝑦 < 𝑧 ) ) → ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
36 |
35
|
reximdva |
⊢ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) → ( ∃ 𝑠 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑠 → ∃ 𝑧 ∈ 𝑆 𝑦 < 𝑧 ) ) → ∃ 𝑠 ∈ 𝑆 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
37 |
|
ssrexv |
⊢ ( 𝑆 ⊆ ℕ0 → ( ∃ 𝑠 ∈ 𝑆 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) → ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
38 |
22 36 37
|
sylsyld |
⊢ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) → ( ∃ 𝑠 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ¬ 𝑠 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑠 → ∃ 𝑧 ∈ 𝑆 𝑦 < 𝑧 ) ) → ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
39 |
21 38
|
mpd |
⊢ ( ( 𝑆 ∈ Fin ∧ 𝑆 ≠ ∅ ∧ 𝑆 ⊆ ℕ0 ) → ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) |
40 |
39
|
3exp |
⊢ ( 𝑆 ∈ Fin → ( 𝑆 ≠ ∅ → ( 𝑆 ⊆ ℕ0 → ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) ) |
41 |
40
|
com3l |
⊢ ( 𝑆 ≠ ∅ → ( 𝑆 ⊆ ℕ0 → ( 𝑆 ∈ Fin → ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) ) |
42 |
14 41
|
pm2.61ine |
⊢ ( 𝑆 ⊆ ℕ0 → ( 𝑆 ∈ Fin → ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |
43 |
|
fzfi |
⊢ ( 0 ... 𝑠 ) ∈ Fin |
44 |
|
elfz2nn0 |
⊢ ( 𝑦 ∈ ( 0 ... 𝑠 ) ↔ ( 𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠 ) ) |
45 |
44
|
notbii |
⊢ ( ¬ 𝑦 ∈ ( 0 ... 𝑠 ) ↔ ¬ ( 𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠 ) ) |
46 |
|
3ianor |
⊢ ( ¬ ( 𝑦 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ∧ 𝑦 ≤ 𝑠 ) ↔ ( ¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠 ) ) |
47 |
|
3orass |
⊢ ( ( ¬ 𝑦 ∈ ℕ0 ∨ ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠 ) ↔ ( ¬ 𝑦 ∈ ℕ0 ∨ ( ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠 ) ) ) |
48 |
45 46 47
|
3bitri |
⊢ ( ¬ 𝑦 ∈ ( 0 ... 𝑠 ) ↔ ( ¬ 𝑦 ∈ ℕ0 ∨ ( ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠 ) ) ) |
49 |
|
ssel |
⊢ ( 𝑆 ⊆ ℕ0 → ( 𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0 ) ) |
50 |
49
|
adantr |
⊢ ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( 𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0 ) ) |
51 |
50
|
adantr |
⊢ ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ( 𝑦 ∈ 𝑆 → 𝑦 ∈ ℕ0 ) ) |
52 |
51
|
con3rr3 |
⊢ ( ¬ 𝑦 ∈ ℕ0 → ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ¬ 𝑦 ∈ 𝑆 ) ) |
53 |
|
notnotb |
⊢ ( 𝑦 ∈ ℕ0 ↔ ¬ ¬ 𝑦 ∈ ℕ0 ) |
54 |
|
pm2.24 |
⊢ ( 𝑠 ∈ ℕ0 → ( ¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆 ) ) |
55 |
54
|
adantl |
⊢ ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆 ) ) |
56 |
55
|
adantr |
⊢ ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ( ¬ 𝑠 ∈ ℕ0 → ¬ 𝑦 ∈ 𝑆 ) ) |
57 |
56
|
com12 |
⊢ ( ¬ 𝑠 ∈ ℕ0 → ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ¬ 𝑦 ∈ 𝑆 ) ) |
58 |
57
|
a1d |
⊢ ( ¬ 𝑠 ∈ ℕ0 → ( 𝑦 ∈ ℕ0 → ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ¬ 𝑦 ∈ 𝑆 ) ) ) |
59 |
|
breq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑠 < 𝑥 ↔ 𝑠 < 𝑦 ) ) |
60 |
|
neleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∉ 𝑆 ↔ 𝑦 ∉ 𝑆 ) ) |
61 |
59 60
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ↔ ( 𝑠 < 𝑦 → 𝑦 ∉ 𝑆 ) ) ) |
62 |
61
|
rspcva |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ( 𝑠 < 𝑦 → 𝑦 ∉ 𝑆 ) ) |
63 |
|
nn0re |
⊢ ( 𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ ) |
64 |
|
nn0re |
⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ ) |
65 |
|
ltnle |
⊢ ( ( 𝑠 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠 ) ) |
66 |
63 64 65
|
syl2an |
⊢ ( ( 𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) → ( 𝑠 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑠 ) ) |
67 |
|
df-nel |
⊢ ( 𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆 ) |
68 |
67
|
a1i |
⊢ ( ( 𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) → ( 𝑦 ∉ 𝑆 ↔ ¬ 𝑦 ∈ 𝑆 ) ) |
69 |
66 68
|
imbi12d |
⊢ ( ( 𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑠 < 𝑦 → 𝑦 ∉ 𝑆 ) ↔ ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) |
70 |
69
|
biimpd |
⊢ ( ( 𝑠 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑠 < 𝑦 → 𝑦 ∉ 𝑆 ) → ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) |
71 |
70
|
ex |
⊢ ( 𝑠 ∈ ℕ0 → ( 𝑦 ∈ ℕ0 → ( ( 𝑠 < 𝑦 → 𝑦 ∉ 𝑆 ) → ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) ) |
72 |
71
|
adantl |
⊢ ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( 𝑦 ∈ ℕ0 → ( ( 𝑠 < 𝑦 → 𝑦 ∉ 𝑆 ) → ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) ) |
73 |
72
|
com12 |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ( 𝑠 < 𝑦 → 𝑦 ∉ 𝑆 ) → ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) ) |
74 |
73
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ( 𝑠 < 𝑦 → 𝑦 ∉ 𝑆 ) → ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) ) |
75 |
62 74
|
mpid |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) |
76 |
75
|
ex |
⊢ ( 𝑦 ∈ ℕ0 → ( ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) → ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) ) |
77 |
76
|
com13 |
⊢ ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) → ( 𝑦 ∈ ℕ0 → ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) ) |
78 |
77
|
imp |
⊢ ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ( 𝑦 ∈ ℕ0 → ( ¬ 𝑦 ≤ 𝑠 → ¬ 𝑦 ∈ 𝑆 ) ) ) |
79 |
78
|
com13 |
⊢ ( ¬ 𝑦 ≤ 𝑠 → ( 𝑦 ∈ ℕ0 → ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ¬ 𝑦 ∈ 𝑆 ) ) ) |
80 |
58 79
|
jaoi |
⊢ ( ( ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠 ) → ( 𝑦 ∈ ℕ0 → ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ¬ 𝑦 ∈ 𝑆 ) ) ) |
81 |
53 80
|
syl5bir |
⊢ ( ( ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠 ) → ( ¬ ¬ 𝑦 ∈ ℕ0 → ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ¬ 𝑦 ∈ 𝑆 ) ) ) |
82 |
81
|
impcom |
⊢ ( ( ¬ ¬ 𝑦 ∈ ℕ0 ∧ ( ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠 ) ) → ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ¬ 𝑦 ∈ 𝑆 ) ) |
83 |
52 82
|
jaoi3 |
⊢ ( ( ¬ 𝑦 ∈ ℕ0 ∨ ( ¬ 𝑠 ∈ ℕ0 ∨ ¬ 𝑦 ≤ 𝑠 ) ) → ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ¬ 𝑦 ∈ 𝑆 ) ) |
84 |
48 83
|
sylbi |
⊢ ( ¬ 𝑦 ∈ ( 0 ... 𝑠 ) → ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ¬ 𝑦 ∈ 𝑆 ) ) |
85 |
84
|
com12 |
⊢ ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ( ¬ 𝑦 ∈ ( 0 ... 𝑠 ) → ¬ 𝑦 ∈ 𝑆 ) ) |
86 |
85
|
con4d |
⊢ ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → ( 𝑦 ∈ 𝑆 → 𝑦 ∈ ( 0 ... 𝑠 ) ) ) |
87 |
86
|
ssrdv |
⊢ ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → 𝑆 ⊆ ( 0 ... 𝑠 ) ) |
88 |
|
ssfi |
⊢ ( ( ( 0 ... 𝑠 ) ∈ Fin ∧ 𝑆 ⊆ ( 0 ... 𝑠 ) ) → 𝑆 ∈ Fin ) |
89 |
43 87 88
|
sylancr |
⊢ ( ( ( 𝑆 ⊆ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) → 𝑆 ∈ Fin ) |
90 |
89
|
rexlimdva2 |
⊢ ( 𝑆 ⊆ ℕ0 → ( ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) → 𝑆 ∈ Fin ) ) |
91 |
42 90
|
impbid |
⊢ ( 𝑆 ⊆ ℕ0 → ( 𝑆 ∈ Fin ↔ ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → 𝑥 ∉ 𝑆 ) ) ) |