Step |
Hyp |
Ref |
Expression |
1 |
|
tosso.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
tosso.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
tosso.s |
⊢ < = ( lt ‘ 𝐾 ) |
4 |
1 2 3
|
pleval2 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ) ) |
5 |
4
|
3expb |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ) ) |
6 |
1 2 3
|
pleval2 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) ) |
7 |
|
equcom |
⊢ ( 𝑦 = 𝑥 ↔ 𝑥 = 𝑦 ) |
8 |
7
|
orbi2i |
⊢ ( ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ↔ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) |
9 |
6 8
|
bitrdi |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
10 |
9
|
3com23 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
11 |
10
|
3expb |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
12 |
5 11
|
orbi12d |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) ) ) |
13 |
|
df-3or |
⊢ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ 𝑦 < 𝑥 ) ) |
14 |
|
or32 |
⊢ ( ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ 𝑦 < 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ) ∨ 𝑥 = 𝑦 ) ) |
15 |
|
orordir |
⊢ ( ( ( 𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ) ∨ 𝑥 = 𝑦 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
16 |
14 15
|
bitri |
⊢ ( ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ 𝑦 < 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
17 |
13 16
|
bitri |
⊢ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
18 |
12 17
|
bitr4di |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ↔ ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) |
19 |
18
|
2ralbidva |
⊢ ( 𝐾 ∈ Poset → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) |
20 |
19
|
pm5.32i |
⊢ ( ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) ↔ ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) |
21 |
1 2 3
|
pospo |
⊢ ( 𝐾 ∈ 𝑉 → ( 𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ) |
22 |
21
|
anbi1d |
⊢ ( 𝐾 ∈ 𝑉 → ( ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ↔ ( ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) ) |
23 |
20 22
|
syl5bb |
⊢ ( 𝐾 ∈ 𝑉 → ( ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) ↔ ( ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) ) |
24 |
1 2
|
istos |
⊢ ( 𝐾 ∈ Toset ↔ ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) ) |
25 |
|
df-so |
⊢ ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) |
26 |
25
|
anbi1i |
⊢ ( ( < Or 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ↔ ( ( < Po 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) |
27 |
|
an32 |
⊢ ( ( ( < Po 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ↔ ( ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) |
28 |
26 27
|
bitri |
⊢ ( ( < Or 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ↔ ( ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) |
29 |
23 24 28
|
3bitr4g |
⊢ ( 𝐾 ∈ 𝑉 → ( 𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) ) |