Step |
Hyp |
Ref |
Expression |
1 |
|
cotr |
⊢ ( ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ↔ ∀ 𝑎 ∀ 𝑏 ∀ 𝑐 ( ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) → 𝑎 𝑟 𝑐 ) ) |
2 |
|
sp |
⊢ ( ∀ 𝑎 ∀ 𝑏 ∀ 𝑐 ( ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) → 𝑎 𝑟 𝑐 ) → ∀ 𝑏 ∀ 𝑐 ( ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) → 𝑎 𝑟 𝑐 ) ) |
3 |
2
|
19.21bbi |
⊢ ( ∀ 𝑎 ∀ 𝑏 ∀ 𝑐 ( ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) → 𝑎 𝑟 𝑐 ) → ( ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) → 𝑎 𝑟 𝑐 ) ) |
4 |
1 3
|
sylbi |
⊢ ( ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 → ( ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) → 𝑎 𝑟 𝑐 ) ) |
5 |
4
|
adantl |
⊢ ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) → 𝑎 𝑟 𝑐 ) ) |
6 |
5
|
a2i |
⊢ ( ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) → ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑐 ) ) |
7 |
6
|
alimi |
⊢ ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) → ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑐 ) ) |
8 |
7
|
ax-gen |
⊢ ∀ 𝑐 ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) → ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑐 ) ) |
9 |
8
|
ax-gen |
⊢ ∀ 𝑏 ∀ 𝑐 ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) → ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑐 ) ) |
10 |
9
|
ax-gen |
⊢ ∀ 𝑎 ∀ 𝑏 ∀ 𝑐 ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) → ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑐 ) ) |
11 |
|
brtrclfv |
⊢ ( 𝑅 ∈ 𝑉 → ( 𝑎 ( t+ ‘ 𝑅 ) 𝑏 ↔ ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑏 ) ) ) |
12 |
|
brtrclfv |
⊢ ( 𝑅 ∈ 𝑉 → ( 𝑏 ( t+ ‘ 𝑅 ) 𝑐 ↔ ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑏 𝑟 𝑐 ) ) ) |
13 |
11 12
|
anbi12d |
⊢ ( 𝑅 ∈ 𝑉 → ( ( 𝑎 ( t+ ‘ 𝑅 ) 𝑏 ∧ 𝑏 ( t+ ‘ 𝑅 ) 𝑐 ) ↔ ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑏 ) ∧ ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑏 𝑟 𝑐 ) ) ) ) |
14 |
|
jcab |
⊢ ( ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) ↔ ( ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑏 ) ∧ ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑏 𝑟 𝑐 ) ) ) |
15 |
14
|
albii |
⊢ ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) ↔ ∀ 𝑟 ( ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑏 ) ∧ ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑏 𝑟 𝑐 ) ) ) |
16 |
|
19.26 |
⊢ ( ∀ 𝑟 ( ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑏 ) ∧ ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑏 𝑟 𝑐 ) ) ↔ ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑏 ) ∧ ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑏 𝑟 𝑐 ) ) ) |
17 |
15 16
|
bitri |
⊢ ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) ↔ ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑏 ) ∧ ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑏 𝑟 𝑐 ) ) ) |
18 |
13 17
|
bitr4di |
⊢ ( 𝑅 ∈ 𝑉 → ( ( 𝑎 ( t+ ‘ 𝑅 ) 𝑏 ∧ 𝑏 ( t+ ‘ 𝑅 ) 𝑐 ) ↔ ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) ) ) |
19 |
|
brtrclfv |
⊢ ( 𝑅 ∈ 𝑉 → ( 𝑎 ( t+ ‘ 𝑅 ) 𝑐 ↔ ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑐 ) ) ) |
20 |
18 19
|
imbi12d |
⊢ ( 𝑅 ∈ 𝑉 → ( ( ( 𝑎 ( t+ ‘ 𝑅 ) 𝑏 ∧ 𝑏 ( t+ ‘ 𝑅 ) 𝑐 ) → 𝑎 ( t+ ‘ 𝑅 ) 𝑐 ) ↔ ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) → ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑐 ) ) ) ) |
21 |
20
|
albidv |
⊢ ( 𝑅 ∈ 𝑉 → ( ∀ 𝑐 ( ( 𝑎 ( t+ ‘ 𝑅 ) 𝑏 ∧ 𝑏 ( t+ ‘ 𝑅 ) 𝑐 ) → 𝑎 ( t+ ‘ 𝑅 ) 𝑐 ) ↔ ∀ 𝑐 ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) → ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑐 ) ) ) ) |
22 |
21
|
2albidv |
⊢ ( 𝑅 ∈ 𝑉 → ( ∀ 𝑎 ∀ 𝑏 ∀ 𝑐 ( ( 𝑎 ( t+ ‘ 𝑅 ) 𝑏 ∧ 𝑏 ( t+ ‘ 𝑅 ) 𝑐 ) → 𝑎 ( t+ ‘ 𝑅 ) 𝑐 ) ↔ ∀ 𝑎 ∀ 𝑏 ∀ 𝑐 ( ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → ( 𝑎 𝑟 𝑏 ∧ 𝑏 𝑟 𝑐 ) ) → ∀ 𝑟 ( ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) → 𝑎 𝑟 𝑐 ) ) ) ) |
23 |
10 22
|
mpbiri |
⊢ ( 𝑅 ∈ 𝑉 → ∀ 𝑎 ∀ 𝑏 ∀ 𝑐 ( ( 𝑎 ( t+ ‘ 𝑅 ) 𝑏 ∧ 𝑏 ( t+ ‘ 𝑅 ) 𝑐 ) → 𝑎 ( t+ ‘ 𝑅 ) 𝑐 ) ) |
24 |
|
cotr |
⊢ ( ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) ↔ ∀ 𝑎 ∀ 𝑏 ∀ 𝑐 ( ( 𝑎 ( t+ ‘ 𝑅 ) 𝑏 ∧ 𝑏 ( t+ ‘ 𝑅 ) 𝑐 ) → 𝑎 ( t+ ‘ 𝑅 ) 𝑐 ) ) |
25 |
23 24
|
sylibr |
⊢ ( 𝑅 ∈ 𝑉 → ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) ) |