Step |
Hyp |
Ref |
Expression |
1 |
|
isso2i.1 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 ↔ ¬ ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) ) |
2 |
|
isso2i.2 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) |
3 |
|
equid |
⊢ 𝑥 = 𝑥 |
4 |
3
|
orci |
⊢ ( 𝑥 = 𝑥 ∨ 𝑥 𝑅 𝑥 ) |
5 |
|
nfv |
⊢ Ⅎ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 = 𝑥 ∨ 𝑥 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑥 ) ) |
6 |
|
eleq1w |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴 ) ) |
7 |
6
|
anbi2d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ) ) |
8 |
|
equequ2 |
⊢ ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑥 ) ) |
9 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 𝑅 𝑥 ↔ 𝑥 𝑅 𝑥 ) ) |
10 |
8 9
|
orbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ↔ ( 𝑥 = 𝑥 ∨ 𝑥 𝑅 𝑥 ) ) ) |
11 |
|
breq2 |
⊢ ( 𝑦 = 𝑥 → ( 𝑥 𝑅 𝑦 ↔ 𝑥 𝑅 𝑥 ) ) |
12 |
11
|
notbid |
⊢ ( 𝑦 = 𝑥 → ( ¬ 𝑥 𝑅 𝑦 ↔ ¬ 𝑥 𝑅 𝑥 ) ) |
13 |
10 12
|
bibi12d |
⊢ ( 𝑦 = 𝑥 → ( ( ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑦 ) ↔ ( ( 𝑥 = 𝑥 ∨ 𝑥 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑥 ) ) ) |
14 |
7 13
|
imbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑦 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 = 𝑥 ∨ 𝑥 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑥 ) ) ) ) |
15 |
1
|
con2bid |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑦 ) ) |
16 |
5 14 15
|
chvarfv |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 = 𝑥 ∨ 𝑥 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑥 ) ) |
17 |
4 16
|
mpbii |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ¬ 𝑥 𝑅 𝑥 ) |
18 |
17
|
anidms |
⊢ ( 𝑥 ∈ 𝐴 → ¬ 𝑥 𝑅 𝑥 ) |
19 |
15
|
biimprd |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( ¬ 𝑥 𝑅 𝑦 → ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) ) |
20 |
19
|
orrd |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 ∨ ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) ) |
21 |
|
3orass |
⊢ ( ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ↔ ( 𝑥 𝑅 𝑦 ∨ ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) ) |
22 |
20 21
|
sylibr |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝑅 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) |
23 |
18 2 22
|
issoi |
⊢ 𝑅 Or 𝐴 |