Step |
Hyp |
Ref |
Expression |
1 |
|
lrrec.1 |
⊢ 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ 𝑥 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) } |
2 |
1
|
lrrecval |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 𝑅 𝐵 ↔ 𝐴 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ) ) |
3 |
|
lrold |
⊢ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) = ( O ‘ ( bday ‘ 𝐵 ) ) |
4 |
3
|
a1i |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) = ( O ‘ ( bday ‘ 𝐵 ) ) ) |
5 |
4
|
eleq2d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ∈ ( ( L ‘ 𝐵 ) ∪ ( R ‘ 𝐵 ) ) ↔ 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ) ) |
6 |
|
bdayelon |
⊢ ( bday ‘ 𝐵 ) ∈ On |
7 |
|
oldbday |
⊢ ( ( ( bday ‘ 𝐵 ) ∈ On ∧ 𝐴 ∈ No ) → ( 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ↔ ( bday ‘ 𝐴 ) ∈ ( bday ‘ 𝐵 ) ) ) |
8 |
6 7
|
mpan |
⊢ ( 𝐴 ∈ No → ( 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ↔ ( bday ‘ 𝐴 ) ∈ ( bday ‘ 𝐵 ) ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ∈ ( O ‘ ( bday ‘ 𝐵 ) ) ↔ ( bday ‘ 𝐴 ) ∈ ( bday ‘ 𝐵 ) ) ) |
10 |
2 5 9
|
3bitrd |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 𝑅 𝐵 ↔ ( bday ‘ 𝐴 ) ∈ ( bday ‘ 𝐵 ) ) ) |