Step |
Hyp |
Ref |
Expression |
1 |
|
tleile.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
tleile.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
simp2 |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
4 |
|
simp3 |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
5 |
1 2
|
istos |
⊢ ( 𝐾 ∈ Toset ↔ ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) ) |
6 |
5
|
simprbi |
⊢ ( 𝐾 ∈ Toset → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) |
7 |
6
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) |
8 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦 ) ) |
9 |
|
breq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋 ) ) |
10 |
8 9
|
orbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ↔ ( 𝑋 ≤ 𝑦 ∨ 𝑦 ≤ 𝑋 ) ) ) |
11 |
|
breq2 |
⊢ ( 𝑦 = 𝑌 → ( 𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌 ) ) |
12 |
|
breq1 |
⊢ ( 𝑦 = 𝑌 → ( 𝑦 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋 ) ) |
13 |
11 12
|
orbi12d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 ≤ 𝑦 ∨ 𝑦 ≤ 𝑋 ) ↔ ( 𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋 ) ) ) |
14 |
10 13
|
rspc2va |
⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) → ( 𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋 ) ) |
15 |
3 4 7 14
|
syl21anc |
⊢ ( ( 𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋 ) ) |