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Theorem ssorduni 4304
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.)
Assertion
Ref Expression
ssorduni  |-  ( A 
C_  On  ->  Ord  U. A )

Proof of Theorem ssorduni
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 3657 . . . . 5  |-  ( x  e.  U. A  <->  E. y  e.  A  x  e.  y )
2 ssel 3019 . . . . . . . . 9  |-  ( A 
C_  On  ->  ( y  e.  A  ->  y  e.  On ) )
3 onelss 4214 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  C_  y ) )
42, 3syl6 33 . . . . . . . 8  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  C_  y ) ) )
5 anc2r 321 . . . . . . . 8  |-  ( ( y  e.  A  -> 
( x  e.  y  ->  x  C_  y
) )  ->  (
y  e.  A  -> 
( x  e.  y  ->  ( x  C_  y  /\  y  e.  A
) ) ) )
64, 5syl 14 . . . . . . 7  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  -> 
( x  C_  y  /\  y  e.  A
) ) ) )
7 ssuni 3675 . . . . . . 7  |-  ( ( x  C_  y  /\  y  e.  A )  ->  x  C_  U. A )
86, 7syl8 70 . . . . . 6  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  C_  U. A ) ) )
98rexlimdv 2488 . . . . 5  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  C_ 
U. A ) )
101, 9syl5bi 150 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  C_  U. A ) )
1110ralrimiv 2445 . . 3  |-  ( A 
C_  On  ->  A. x  e.  U. A x  C_  U. A )
12 dftr3 3940 . . 3  |-  ( Tr 
U. A  <->  A. x  e.  U. A x  C_  U. A )
1311, 12sylibr 132 . 2  |-  ( A 
C_  On  ->  Tr  U. A )
14 onelon 4211 . . . . . . 7  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
1514ex 113 . . . . . 6  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  e.  On )
)
162, 15syl6 33 . . . . 5  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  e.  On )
) )
1716rexlimdv 2488 . . . 4  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  e.  On ) )
181, 17syl5bi 150 . . 3  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  e.  On )
)
1918ssrdv 3031 . 2  |-  ( A 
C_  On  ->  U. A  C_  On )
20 ordon 4303 . . 3  |-  Ord  On
21 trssord 4207 . . . 4  |-  ( ( Tr  U. A  /\  U. A  C_  On  /\  Ord  On )  ->  Ord  U. A
)
22213exp 1142 . . 3  |-  ( Tr 
U. A  ->  ( U. A  C_  On  ->  ( Ord  On  ->  Ord  U. A ) ) )
2320, 22mpii 43 . 2  |-  ( Tr 
U. A  ->  ( U. A  C_  On  ->  Ord  U. A ) )
2413, 19, 23sylc 61 1  |-  ( A 
C_  On  ->  Ord  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   A.wral 2359   E.wrex 2360    C_ wss 2999   U.cuni 3653   Tr wtr 3936   Ord word 4189   Oncon0 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195
This theorem is referenced by:  ssonuni  4305  orduni  4312  tfrlem8  6083  tfrexlem  6099
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