Step |
Hyp |
Ref |
Expression |
1 |
|
fimax2g |
⊢ ( ( 𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 𝑅 𝑦 ) |
2 |
|
df-ne |
⊢ ( 𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦 ) |
3 |
2
|
imbi1i |
⊢ ( ( 𝑥 ≠ 𝑦 → 𝑦 𝑅 𝑥 ) ↔ ( ¬ 𝑥 = 𝑦 → 𝑦 𝑅 𝑥 ) ) |
4 |
|
pm4.64 |
⊢ ( ( ¬ 𝑥 = 𝑦 → 𝑦 𝑅 𝑥 ) ↔ ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) |
5 |
3 4
|
bitri |
⊢ ( ( 𝑥 ≠ 𝑦 → 𝑦 𝑅 𝑥 ) ↔ ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) |
6 |
|
sotric |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 𝑅 𝑦 ↔ ¬ ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ) ) |
7 |
6
|
con2bid |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝑥 = 𝑦 ∨ 𝑦 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑦 ) ) |
8 |
5 7
|
syl5bb |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝑥 ≠ 𝑦 → 𝑦 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑦 ) ) |
9 |
8
|
anassrs |
⊢ ( ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑥 ≠ 𝑦 → 𝑦 𝑅 𝑥 ) ↔ ¬ 𝑥 𝑅 𝑦 ) ) |
10 |
9
|
ralbidva |
⊢ ( ( 𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → 𝑦 𝑅 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 𝑅 𝑦 ) ) |
11 |
10
|
rexbidva |
⊢ ( 𝑅 Or 𝐴 → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → 𝑦 𝑅 𝑥 ) ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 𝑅 𝑦 ) ) |
12 |
11
|
3ad2ant1 |
⊢ ( ( 𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → 𝑦 𝑅 𝑥 ) ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 𝑅 𝑦 ) ) |
13 |
1 12
|
mpbird |
⊢ ( ( 𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≠ 𝑦 → 𝑦 𝑅 𝑥 ) ) |