Step |
Hyp |
Ref |
Expression |
1 |
|
elfvdm |
⊢ ( 𝑋 ∈ ( O ‘ 𝐴 ) → 𝐴 ∈ dom O ) |
2 |
|
oldf |
⊢ O : On ⟶ 𝒫 No |
3 |
2
|
fdmi |
⊢ dom O = On |
4 |
1 3
|
eleqtrdi |
⊢ ( 𝑋 ∈ ( O ‘ 𝐴 ) → 𝐴 ∈ On ) |
5 |
|
elold |
⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( O ‘ 𝐴 ) ↔ ∃ 𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) ) |
6 |
|
madebdayim |
⊢ ( 𝑋 ∈ ( M ‘ 𝑏 ) → ( bday ‘ 𝑋 ) ⊆ 𝑏 ) |
7 |
6
|
ad2antll |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘ 𝑏 ) ) ) → ( bday ‘ 𝑋 ) ⊆ 𝑏 ) |
8 |
|
simprl |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘ 𝑏 ) ) ) → 𝑏 ∈ 𝐴 ) |
9 |
|
bdayelon |
⊢ ( bday ‘ 𝑋 ) ∈ On |
10 |
|
ontr2 |
⊢ ( ( ( bday ‘ 𝑋 ) ∈ On ∧ 𝐴 ∈ On ) → ( ( ( bday ‘ 𝑋 ) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴 ) → ( bday ‘ 𝑋 ) ∈ 𝐴 ) ) |
11 |
9 10
|
mpan |
⊢ ( 𝐴 ∈ On → ( ( ( bday ‘ 𝑋 ) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴 ) → ( bday ‘ 𝑋 ) ∈ 𝐴 ) ) |
12 |
11
|
adantr |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘ 𝑏 ) ) ) → ( ( ( bday ‘ 𝑋 ) ⊆ 𝑏 ∧ 𝑏 ∈ 𝐴 ) → ( bday ‘ 𝑋 ) ∈ 𝐴 ) ) |
13 |
7 8 12
|
mp2and |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑋 ∈ ( M ‘ 𝑏 ) ) ) → ( bday ‘ 𝑋 ) ∈ 𝐴 ) |
14 |
13
|
rexlimdvaa |
⊢ ( 𝐴 ∈ On → ( ∃ 𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘ 𝑏 ) → ( bday ‘ 𝑋 ) ∈ 𝐴 ) ) |
15 |
5 14
|
sylbid |
⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( O ‘ 𝐴 ) → ( bday ‘ 𝑋 ) ∈ 𝐴 ) ) |
16 |
4 15
|
mpcom |
⊢ ( 𝑋 ∈ ( O ‘ 𝐴 ) → ( bday ‘ 𝑋 ) ∈ 𝐴 ) |