Step |
Hyp |
Ref |
Expression |
1 |
|
isprs.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
isprs.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
1 2
|
isprs |
⊢ ( 𝐾 ∈ Proset ↔ ( 𝐾 ∈ V ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑥 ≤ 𝑥 ∧ ( ( 𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) ) |
4 |
3
|
simprbi |
⊢ ( 𝐾 ∈ Proset → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑥 ≤ 𝑥 ∧ ( ( 𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) |
5 |
|
breq12 |
⊢ ( ( 𝑥 = 𝑋 ∧ 𝑥 = 𝑋 ) → ( 𝑥 ≤ 𝑥 ↔ 𝑋 ≤ 𝑋 ) ) |
6 |
5
|
anidms |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ 𝑥 ↔ 𝑋 ≤ 𝑋 ) ) |
7 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦 ) ) |
8 |
7
|
anbi1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) ↔ ( 𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) ) ) |
9 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧 ) ) |
10 |
8 9
|
imbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( ( 𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ↔ ( ( 𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑋 ≤ 𝑧 ) ) ) |
11 |
6 10
|
anbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ≤ 𝑥 ∧ ( ( 𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ↔ ( 𝑋 ≤ 𝑋 ∧ ( ( 𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑋 ≤ 𝑧 ) ) ) ) |
12 |
|
breq2 |
⊢ ( 𝑦 = 𝑌 → ( 𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌 ) ) |
13 |
|
breq1 |
⊢ ( 𝑦 = 𝑌 → ( 𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧 ) ) |
14 |
12 13
|
anbi12d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧 ) ) ) |
15 |
14
|
imbi1d |
⊢ ( 𝑦 = 𝑌 → ( ( ( 𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑋 ≤ 𝑧 ) ↔ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧 ) → 𝑋 ≤ 𝑧 ) ) ) |
16 |
15
|
anbi2d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 ≤ 𝑋 ∧ ( ( 𝑋 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑋 ≤ 𝑧 ) ) ↔ ( 𝑋 ≤ 𝑋 ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧 ) → 𝑋 ≤ 𝑧 ) ) ) ) |
17 |
|
breq2 |
⊢ ( 𝑧 = 𝑍 → ( 𝑌 ≤ 𝑧 ↔ 𝑌 ≤ 𝑍 ) ) |
18 |
17
|
anbi2d |
⊢ ( 𝑧 = 𝑍 → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧 ) ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍 ) ) ) |
19 |
|
breq2 |
⊢ ( 𝑧 = 𝑍 → ( 𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍 ) ) |
20 |
18 19
|
imbi12d |
⊢ ( 𝑧 = 𝑍 → ( ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧 ) → 𝑋 ≤ 𝑧 ) ↔ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍 ) → 𝑋 ≤ 𝑍 ) ) ) |
21 |
20
|
anbi2d |
⊢ ( 𝑧 = 𝑍 → ( ( 𝑋 ≤ 𝑋 ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑧 ) → 𝑋 ≤ 𝑧 ) ) ↔ ( 𝑋 ≤ 𝑋 ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍 ) → 𝑋 ≤ 𝑍 ) ) ) ) |
22 |
11 16 21
|
rspc3v |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑥 ≤ 𝑥 ∧ ( ( 𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) → ( 𝑋 ≤ 𝑋 ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍 ) → 𝑋 ≤ 𝑍 ) ) ) ) |
23 |
4 22
|
mpan9 |
⊢ ( ( 𝐾 ∈ Proset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑋 ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍 ) → 𝑋 ≤ 𝑍 ) ) ) |