Metamath Proof Explorer


Theorem oldbdayim

Description: If X is in the old set for A , then the birthday of X is less than A . (Contributed by Scott Fenton, 10-Aug-2024)

Ref Expression
Assertion oldbdayim ( 𝑋 ∈ ( O ‘ 𝐴 ) → ( bday 𝑋 ) ∈ 𝐴 )

Proof

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 ) ↔ ∃ 𝑏𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) )
6 madebdayim ( 𝑋 ∈ ( M ‘ 𝑏 ) → ( bday 𝑋 ) ⊆ 𝑏 )
7 6 ad2antll ( ( 𝐴 ∈ On ∧ ( 𝑏𝐴𝑋 ∈ ( M ‘ 𝑏 ) ) ) → ( bday 𝑋 ) ⊆ 𝑏 )
8 simprl ( ( 𝐴 ∈ On ∧ ( 𝑏𝐴𝑋 ∈ ( M ‘ 𝑏 ) ) ) → 𝑏𝐴 )
9 bdayelon ( bday 𝑋 ) ∈ On
10 ontr2 ( ( ( bday 𝑋 ) ∈ On ∧ 𝐴 ∈ On ) → ( ( ( bday 𝑋 ) ⊆ 𝑏𝑏𝐴 ) → ( bday 𝑋 ) ∈ 𝐴 ) )
11 9 10 mpan ( 𝐴 ∈ On → ( ( ( bday 𝑋 ) ⊆ 𝑏𝑏𝐴 ) → ( bday 𝑋 ) ∈ 𝐴 ) )
12 11 adantr ( ( 𝐴 ∈ On ∧ ( 𝑏𝐴𝑋 ∈ ( M ‘ 𝑏 ) ) ) → ( ( ( bday 𝑋 ) ⊆ 𝑏𝑏𝐴 ) → ( bday 𝑋 ) ∈ 𝐴 ) )
13 7 8 12 mp2and ( ( 𝐴 ∈ On ∧ ( 𝑏𝐴𝑋 ∈ ( M ‘ 𝑏 ) ) ) → ( bday 𝑋 ) ∈ 𝐴 )
14 13 rexlimdvaa ( 𝐴 ∈ On → ( ∃ 𝑏𝐴 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 )