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Metamath Proof Explorer

Theorem oawordeu

Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of TakeutiZaring p. 59. (Contributed by NM, 11-Dec-2004)

Ref Expression
Assertion oawordeu 
|- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> E! x e. On ( A +o x ) = B )

Proof

Step Hyp Ref Expression
1 sseq1 
 |-  ( A = if ( A e. On , A , (/) ) -> ( A C_ B <-> if ( A e. On , A , (/) ) C_ B ) )
2 oveq1 
 |-  ( A = if ( A e. On , A , (/) ) -> ( A +o x ) = ( if ( A e. On , A , (/) ) +o x ) )
3 2 eqeq1d 
 |-  ( A = if ( A e. On , A , (/) ) -> ( ( A +o x ) = B <-> ( if ( A e. On , A , (/) ) +o x ) = B ) )
4 3 reubidv 
 |-  ( A = if ( A e. On , A , (/) ) -> ( E! x e. On ( A +o x ) = B <-> E! x e. On ( if ( A e. On , A , (/) ) +o x ) = B ) )
5 1 4 imbi12d 
 |-  ( A = if ( A e. On , A , (/) ) -> ( ( A C_ B -> E! x e. On ( A +o x ) = B ) <-> ( if ( A e. On , A , (/) ) C_ B -> E! x e. On ( if ( A e. On , A , (/) ) +o x ) = B ) ) )
6 sseq2 
 |-  ( B = if ( B e. On , B , (/) ) -> ( if ( A e. On , A , (/) ) C_ B <-> if ( A e. On , A , (/) ) C_ if ( B e. On , B , (/) ) ) )
7 eqeq2 
 |-  ( B = if ( B e. On , B , (/) ) -> ( ( if ( A e. On , A , (/) ) +o x ) = B <-> ( if ( A e. On , A , (/) ) +o x ) = if ( B e. On , B , (/) ) ) )
8 7 reubidv 
 |-  ( B = if ( B e. On , B , (/) ) -> ( E! x e. On ( if ( A e. On , A , (/) ) +o x ) = B <-> E! x e. On ( if ( A e. On , A , (/) ) +o x ) = if ( B e. On , B , (/) ) ) )
9 6 8 imbi12d 
 |-  ( B = if ( B e. On , B , (/) ) -> ( ( if ( A e. On , A , (/) ) C_ B -> E! x e. On ( if ( A e. On , A , (/) ) +o x ) = B ) <-> ( if ( A e. On , A , (/) ) C_ if ( B e. On , B , (/) ) -> E! x e. On ( if ( A e. On , A , (/) ) +o x ) = if ( B e. On , B , (/) ) ) ) )
10 0elon 
 |-  (/) e. On
11 10 elimel 
 |-  if ( A e. On , A , (/) ) e. On
12 10 elimel 
 |-  if ( B e. On , B , (/) ) e. On
13 eqid 
 |-  { y e. On | if ( B e. On , B , (/) ) C_ ( if ( A e. On , A , (/) ) +o y ) } = { y e. On | if ( B e. On , B , (/) ) C_ ( if ( A e. On , A , (/) ) +o y ) }
14 11 12 13 oawordeulem 
 |-  ( if ( A e. On , A , (/) ) C_ if ( B e. On , B , (/) ) -> E! x e. On ( if ( A e. On , A , (/) ) +o x ) = if ( B e. On , B , (/) ) )
15 5 9 14 dedth2h 
 |-  ( ( A e. On /\ B e. On ) -> ( A C_ B -> E! x e. On ( A +o x ) = B ) )
16 15 imp 
 |-  ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> E! x e. On ( A +o x ) = B )