Step |
Hyp |
Ref |
Expression |
1 |
|
sonr |
⊢ ( ( 𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ¬ 𝐵 𝑅 𝐵 ) |
2 |
|
breq2 |
⊢ ( 𝐵 = 𝐶 → ( 𝐵 𝑅 𝐵 ↔ 𝐵 𝑅 𝐶 ) ) |
3 |
2
|
notbid |
⊢ ( 𝐵 = 𝐶 → ( ¬ 𝐵 𝑅 𝐵 ↔ ¬ 𝐵 𝑅 𝐶 ) ) |
4 |
1 3
|
syl5ibcom |
⊢ ( ( 𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝐵 = 𝐶 → ¬ 𝐵 𝑅 𝐶 ) ) |
5 |
4
|
adantrr |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 = 𝐶 → ¬ 𝐵 𝑅 𝐶 ) ) |
6 |
|
so2nr |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ¬ ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐵 ) ) |
7 |
|
imnan |
⊢ ( ( 𝐵 𝑅 𝐶 → ¬ 𝐶 𝑅 𝐵 ) ↔ ¬ ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐵 ) ) |
8 |
6 7
|
sylibr |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 𝑅 𝐶 → ¬ 𝐶 𝑅 𝐵 ) ) |
9 |
8
|
con2d |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐶 𝑅 𝐵 → ¬ 𝐵 𝑅 𝐶 ) ) |
10 |
5 9
|
jaod |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( ( 𝐵 = 𝐶 ∨ 𝐶 𝑅 𝐵 ) → ¬ 𝐵 𝑅 𝐶 ) ) |
11 |
|
solin |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 𝑅 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 𝑅 𝐵 ) ) |
12 |
|
3orass |
⊢ ( ( 𝐵 𝑅 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 𝑅 𝐵 ) ↔ ( 𝐵 𝑅 𝐶 ∨ ( 𝐵 = 𝐶 ∨ 𝐶 𝑅 𝐵 ) ) ) |
13 |
11 12
|
sylib |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 𝑅 𝐶 ∨ ( 𝐵 = 𝐶 ∨ 𝐶 𝑅 𝐵 ) ) ) |
14 |
13
|
ord |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( ¬ 𝐵 𝑅 𝐶 → ( 𝐵 = 𝐶 ∨ 𝐶 𝑅 𝐵 ) ) ) |
15 |
10 14
|
impbid |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( ( 𝐵 = 𝐶 ∨ 𝐶 𝑅 𝐵 ) ↔ ¬ 𝐵 𝑅 𝐶 ) ) |
16 |
15
|
con2bid |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 𝑅 𝐶 ↔ ¬ ( 𝐵 = 𝐶 ∨ 𝐶 𝑅 𝐵 ) ) ) |