Metamath Proof Explorer

Theorem ordun

Description: The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of TakeutiZaring p. 40. (Contributed by NM, 28-Nov-2003)

Ref Expression
Assertion ordun 
|- ( ( Ord A /\ Ord B ) -> Ord ( A u. B ) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( A u. B ) = ( A u. B )
2 ordequn 
 |-  ( ( Ord A /\ Ord B ) -> ( ( A u. B ) = ( A u. B ) -> ( ( A u. B ) = A \/ ( A u. B ) = B ) ) )
3 1 2 mpi 
 |-  ( ( Ord A /\ Ord B ) -> ( ( A u. B ) = A \/ ( A u. B ) = B ) )
4 ordeq 
 |-  ( ( A u. B ) = A -> ( Ord ( A u. B ) <-> Ord A ) )
5 4 biimprcd 
 |-  ( Ord A -> ( ( A u. B ) = A -> Ord ( A u. B ) ) )
6 ordeq 
 |-  ( ( A u. B ) = B -> ( Ord ( A u. B ) <-> Ord B ) )
7 6 biimprcd 
 |-  ( Ord B -> ( ( A u. B ) = B -> Ord ( A u. B ) ) )
8 5 7 jaao 
 |-  ( ( Ord A /\ Ord B ) -> ( ( ( A u. B ) = A \/ ( A u. B ) = B ) -> Ord ( A u. B ) ) )
9 3 8 mpd 
 |-  ( ( Ord A /\ Ord B ) -> Ord ( A u. B ) )