Step |
Hyp |
Ref |
Expression |
1 |
|
ssel2 |
⊢ ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ ( 0 [,] +∞ ) ) |
2 |
|
0xr |
⊢ 0 ∈ ℝ* |
3 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
4 |
|
iccgelb |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) → 0 ≤ 𝑦 ) |
5 |
2 3 4
|
mp3an12 |
⊢ ( 𝑦 ∈ ( 0 [,] +∞ ) → 0 ≤ 𝑦 ) |
6 |
|
eliccxr |
⊢ ( 𝑦 ∈ ( 0 [,] +∞ ) → 𝑦 ∈ ℝ* ) |
7 |
|
xrlenlt |
⊢ ( ( 0 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 0 ≤ 𝑦 ↔ ¬ 𝑦 < 0 ) ) |
8 |
2 6 7
|
sylancr |
⊢ ( 𝑦 ∈ ( 0 [,] +∞ ) → ( 0 ≤ 𝑦 ↔ ¬ 𝑦 < 0 ) ) |
9 |
5 8
|
mpbid |
⊢ ( 𝑦 ∈ ( 0 [,] +∞ ) → ¬ 𝑦 < 0 ) |
10 |
1 9
|
syl |
⊢ ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑦 ∈ 𝐴 ) → ¬ 𝑦 < 0 ) |
11 |
10
|
ralrimiva |
⊢ ( 𝐴 ⊆ ( 0 [,] +∞ ) → ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 0 ) |
12 |
11
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ∧ 𝑤 ≤ 0 ) → ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 0 ) |
13 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
14 |
|
ssralv |
⊢ ( ( 0 [,] +∞ ) ⊆ ℝ* → ( ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) → ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
15 |
13 14
|
ax-mp |
⊢ ( ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) → ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) |
16 |
|
simplll |
⊢ ( ( ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 0 < 𝑦 ) → 𝑤 ∈ ℝ* ) |
17 |
2
|
a1i |
⊢ ( ( ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 0 < 𝑦 ) → 0 ∈ ℝ* ) |
18 |
|
simplr |
⊢ ( ( ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 0 < 𝑦 ) → 𝑦 ∈ ( 0 [,] +∞ ) ) |
19 |
13 18
|
sseldi |
⊢ ( ( ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 0 < 𝑦 ) → 𝑦 ∈ ℝ* ) |
20 |
|
simpllr |
⊢ ( ( ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 0 < 𝑦 ) → 𝑤 ≤ 0 ) |
21 |
|
simpr |
⊢ ( ( ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 0 < 𝑦 ) → 0 < 𝑦 ) |
22 |
16 17 19 20 21
|
xrlelttrd |
⊢ ( ( ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) ∧ 0 < 𝑦 ) → 𝑤 < 𝑦 ) |
23 |
22
|
ex |
⊢ ( ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) → ( 0 < 𝑦 → 𝑤 < 𝑦 ) ) |
24 |
23
|
imim1d |
⊢ ( ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) → ( ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) → ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
25 |
24
|
ralimdva |
⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) → ( ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) → ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
26 |
15 25
|
syl5 |
⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝑤 ≤ 0 ) → ( ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) → ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
27 |
26
|
adantll |
⊢ ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ 𝑤 ≤ 0 ) → ( ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) → ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
28 |
27
|
imp |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ 𝑤 ≤ 0 ) ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) → ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) |
29 |
28
|
adantrl |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ 𝑤 ≤ 0 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) → ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) |
30 |
29
|
an32s |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ∧ 𝑤 ≤ 0 ) → ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) |
31 |
|
0e0iccpnf |
⊢ 0 ∈ ( 0 [,] +∞ ) |
32 |
|
breq2 |
⊢ ( 𝑥 = 0 → ( 𝑦 < 𝑥 ↔ 𝑦 < 0 ) ) |
33 |
32
|
notbid |
⊢ ( 𝑥 = 0 → ( ¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 0 ) ) |
34 |
33
|
ralbidv |
⊢ ( 𝑥 = 0 → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 0 ) ) |
35 |
|
breq1 |
⊢ ( 𝑥 = 0 → ( 𝑥 < 𝑦 ↔ 0 < 𝑦 ) ) |
36 |
35
|
imbi1d |
⊢ ( 𝑥 = 0 → ( ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ↔ ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
37 |
36
|
ralbidv |
⊢ ( 𝑥 = 0 → ( ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ↔ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
38 |
34 37
|
anbi12d |
⊢ ( 𝑥 = 0 → ( ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 0 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ) |
39 |
38
|
rspcev |
⊢ ( ( 0 ∈ ( 0 [,] +∞ ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 0 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
40 |
31 39
|
mpan |
⊢ ( ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 0 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 0 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
41 |
12 30 40
|
syl2anc |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ∧ 𝑤 ≤ 0 ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
42 |
|
simpllr |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ∧ 0 ≤ 𝑤 ) → 𝑤 ∈ ℝ* ) |
43 |
|
simpr |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ∧ 0 ≤ 𝑤 ) → 0 ≤ 𝑤 ) |
44 |
|
elxrge0 |
⊢ ( 𝑤 ∈ ( 0 [,] +∞ ) ↔ ( 𝑤 ∈ ℝ* ∧ 0 ≤ 𝑤 ) ) |
45 |
42 43 44
|
sylanbrc |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ∧ 0 ≤ 𝑤 ) → 𝑤 ∈ ( 0 [,] +∞ ) ) |
46 |
15
|
a1i |
⊢ ( 𝐴 ⊆ ( 0 [,] +∞ ) → ( ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) → ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
47 |
46
|
anim2d |
⊢ ( 𝐴 ⊆ ( 0 [,] +∞ ) → ( ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ) |
48 |
47
|
adantr |
⊢ ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) → ( ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ) |
49 |
48
|
imp |
⊢ ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
50 |
49
|
adantr |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ∧ 0 ≤ 𝑤 ) → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
51 |
|
breq2 |
⊢ ( 𝑥 = 𝑤 → ( 𝑦 < 𝑥 ↔ 𝑦 < 𝑤 ) ) |
52 |
51
|
notbid |
⊢ ( 𝑥 = 𝑤 → ( ¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < 𝑤 ) ) |
53 |
52
|
ralbidv |
⊢ ( 𝑥 = 𝑤 → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ) ) |
54 |
|
breq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 < 𝑦 ↔ 𝑤 < 𝑦 ) ) |
55 |
54
|
imbi1d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ↔ ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
56 |
55
|
ralbidv |
⊢ ( 𝑥 = 𝑤 → ( ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ↔ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
57 |
53 56
|
anbi12d |
⊢ ( 𝑥 = 𝑤 → ( ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ) |
58 |
57
|
rspcev |
⊢ ( ( 𝑤 ∈ ( 0 [,] +∞ ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
59 |
45 50 58
|
syl2anc |
⊢ ( ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) ∧ 0 ≤ 𝑤 ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
60 |
|
simplr |
⊢ ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) → 𝑤 ∈ ℝ* ) |
61 |
2
|
a1i |
⊢ ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) → 0 ∈ ℝ* ) |
62 |
|
xrletri |
⊢ ( ( 𝑤 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝑤 ≤ 0 ∨ 0 ≤ 𝑤 ) ) |
63 |
60 61 62
|
syl2anc |
⊢ ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) → ( 𝑤 ≤ 0 ∨ 0 ≤ 𝑤 ) ) |
64 |
41 59 63
|
mpjaodan |
⊢ ( ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ 𝑤 ∈ ℝ* ) ∧ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
65 |
|
sstr |
⊢ ( ( 𝐴 ⊆ ( 0 [,] +∞ ) ∧ ( 0 [,] +∞ ) ⊆ ℝ* ) → 𝐴 ⊆ ℝ* ) |
66 |
13 65
|
mpan2 |
⊢ ( 𝐴 ⊆ ( 0 [,] +∞ ) → 𝐴 ⊆ ℝ* ) |
67 |
|
xrinfmss |
⊢ ( 𝐴 ⊆ ℝ* → ∃ 𝑤 ∈ ℝ* ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
68 |
66 67
|
syl |
⊢ ( 𝐴 ⊆ ( 0 [,] +∞ ) → ∃ 𝑤 ∈ ℝ* ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑤 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑤 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
69 |
64 68
|
r19.29a |
⊢ ( 𝐴 ⊆ ( 0 [,] +∞ ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |