Metamath Proof Explorer


Theorem nnuni

Description: The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024)

Ref Expression
Assertion nnuni AωAω

Proof

Step Hyp Ref Expression
1 nn0suc AωA=xωA=sucx
2 unieq A=A=
3 uni0 =
4 2 3 eqtrdi A=A=
5 peano1 ω
6 4 5 eqeltrdi A=Aω
7 nnord xωOrdx
8 ordunisuc Ordxsucx=x
9 7 8 syl xωsucx=x
10 id xωxω
11 9 10 eqeltrd xωsucxω
12 unieq A=sucxA=sucx
13 12 eleq1d A=sucxAωsucxω
14 11 13 syl5ibrcom xωA=sucxAω
15 14 rexlimiv xωA=sucxAω
16 6 15 jaoi A=xωA=sucxAω
17 1 16 syl AωAω