Step |
Hyp |
Ref |
Expression |
1 |
|
supssd.0 |
⊢ ( 𝜑 → 𝑅 Or 𝐴 ) |
2 |
|
supssd.1 |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 ) |
3 |
|
supssd.2 |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 ) |
4 |
|
supssd.3 |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) |
5 |
|
supssd.4 |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐶 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐶 𝑦 𝑅 𝑧 ) ) ) |
6 |
1 5
|
supcl |
⊢ ( 𝜑 → sup ( 𝐶 , 𝐴 , 𝑅 ) ∈ 𝐴 ) |
7 |
2
|
sseld |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐶 ) ) |
8 |
1 5
|
supub |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐶 → ¬ sup ( 𝐶 , 𝐴 , 𝑅 ) 𝑅 𝑧 ) ) |
9 |
7 8
|
syld |
⊢ ( 𝜑 → ( 𝑧 ∈ 𝐵 → ¬ sup ( 𝐶 , 𝐴 , 𝑅 ) 𝑅 𝑧 ) ) |
10 |
9
|
ralrimiv |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ¬ sup ( 𝐶 , 𝐴 , 𝑅 ) 𝑅 𝑧 ) |
11 |
1 4
|
supnub |
⊢ ( 𝜑 → ( ( sup ( 𝐶 , 𝐴 , 𝑅 ) ∈ 𝐴 ∧ ∀ 𝑧 ∈ 𝐵 ¬ sup ( 𝐶 , 𝐴 , 𝑅 ) 𝑅 𝑧 ) → ¬ sup ( 𝐶 , 𝐴 , 𝑅 ) 𝑅 sup ( 𝐵 , 𝐴 , 𝑅 ) ) ) |
12 |
6 10 11
|
mp2and |
⊢ ( 𝜑 → ¬ sup ( 𝐶 , 𝐴 , 𝑅 ) 𝑅 sup ( 𝐵 , 𝐴 , 𝑅 ) ) |