Metamath Proof Explorer


Theorem ssorduni

Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of TakeutiZaring p. 40. (Contributed by NM, 30-May-1994) (Proof shortened by Andrew Salmon, 12-Aug-2011)

Ref Expression
Assertion ssorduni ( 𝐴 ⊆ On → Ord 𝐴 )

Proof

Step Hyp Ref Expression
1 eluni2 ( 𝑥 𝐴 ↔ ∃ 𝑦𝐴 𝑥𝑦 )
2 ssel ( 𝐴 ⊆ On → ( 𝑦𝐴𝑦 ∈ On ) )
3 onelss ( 𝑦 ∈ On → ( 𝑥𝑦𝑥𝑦 ) )
4 2 3 syl6 ( 𝐴 ⊆ On → ( 𝑦𝐴 → ( 𝑥𝑦𝑥𝑦 ) ) )
5 anc2r ( ( 𝑦𝐴 → ( 𝑥𝑦𝑥𝑦 ) ) → ( 𝑦𝐴 → ( 𝑥𝑦 → ( 𝑥𝑦𝑦𝐴 ) ) ) )
6 4 5 syl ( 𝐴 ⊆ On → ( 𝑦𝐴 → ( 𝑥𝑦 → ( 𝑥𝑦𝑦𝐴 ) ) ) )
7 ssuni ( ( 𝑥𝑦𝑦𝐴 ) → 𝑥 𝐴 )
8 6 7 syl8 ( 𝐴 ⊆ On → ( 𝑦𝐴 → ( 𝑥𝑦𝑥 𝐴 ) ) )
9 8 rexlimdv ( 𝐴 ⊆ On → ( ∃ 𝑦𝐴 𝑥𝑦𝑥 𝐴 ) )
10 1 9 syl5bi ( 𝐴 ⊆ On → ( 𝑥 𝐴𝑥 𝐴 ) )
11 10 ralrimiv ( 𝐴 ⊆ On → ∀ 𝑥 𝐴 𝑥 𝐴 )
12 dftr3 ( Tr 𝐴 ↔ ∀ 𝑥 𝐴 𝑥 𝐴 )
13 11 12 sylibr ( 𝐴 ⊆ On → Tr 𝐴 )
14 onelon ( ( 𝑦 ∈ On ∧ 𝑥𝑦 ) → 𝑥 ∈ On )
15 14 ex ( 𝑦 ∈ On → ( 𝑥𝑦𝑥 ∈ On ) )
16 2 15 syl6 ( 𝐴 ⊆ On → ( 𝑦𝐴 → ( 𝑥𝑦𝑥 ∈ On ) ) )
17 16 rexlimdv ( 𝐴 ⊆ On → ( ∃ 𝑦𝐴 𝑥𝑦𝑥 ∈ On ) )
18 1 17 syl5bi ( 𝐴 ⊆ On → ( 𝑥 𝐴𝑥 ∈ On ) )
19 18 ssrdv ( 𝐴 ⊆ On → 𝐴 ⊆ On )
20 ordon Ord On
21 trssord ( ( Tr 𝐴 𝐴 ⊆ On ∧ Ord On ) → Ord 𝐴 )
22 21 3exp ( Tr 𝐴 → ( 𝐴 ⊆ On → ( Ord On → Ord 𝐴 ) ) )
23 20 22 mpii ( Tr 𝐴 → ( 𝐴 ⊆ On → Ord 𝐴 ) )
24 13 19 23 sylc ( 𝐴 ⊆ On → Ord 𝐴 )