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Theorem ssorduni 4758
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 4011 . . . . 5  |-  ( x  e.  U. A  <->  E. y  e.  A  x  e.  y )
2 ssel 3334 . . . . . . . . 9  |-  ( A 
C_  On  ->  ( y  e.  A  ->  y  e.  On ) )
3 onelss 4615 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  C_  y ) )
42, 3syl6 31 . . . . . . . 8  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  C_  y ) ) )
5 anc2r 540 . . . . . . . 8  |-  ( ( y  e.  A  -> 
( x  e.  y  ->  x  C_  y
) )  ->  (
y  e.  A  -> 
( x  e.  y  ->  ( x  C_  y  /\  y  e.  A
) ) ) )
64, 5syl 16 . . . . . . 7  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  -> 
( x  C_  y  /\  y  e.  A
) ) ) )
7 ssuni 4029 . . . . . . 7  |-  ( ( x  C_  y  /\  y  e.  A )  ->  x  C_  U. A )
86, 7syl8 67 . . . . . 6  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  C_  U. A ) ) )
98rexlimdv 2821 . . . . 5  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  C_ 
U. A ) )
101, 9syl5bi 209 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  C_  U. A ) )
1110ralrimiv 2780 . . 3  |-  ( A 
C_  On  ->  A. x  e.  U. A x  C_  U. A )
12 dftr3 4298 . . 3  |-  ( Tr 
U. A  <->  A. x  e.  U. A x  C_  U. A )
1311, 12sylibr 204 . 2  |-  ( A 
C_  On  ->  Tr  U. A )
14 onelon 4598 . . . . . . 7  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
1514ex 424 . . . . . 6  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  e.  On )
)
162, 15syl6 31 . . . . 5  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  e.  On )
) )
1716rexlimdv 2821 . . . 4  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  e.  On ) )
181, 17syl5bi 209 . . 3  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  e.  On )
)
1918ssrdv 3346 . 2  |-  ( A 
C_  On  ->  U. A  C_  On )
20 ordon 4755 . . 3  |-  Ord  On
21 trssord 4590 . . . 4  |-  ( ( Tr  U. A  /\  U. A  C_  On  /\  Ord  On )  ->  Ord  U. A
)
22213exp 1152 . . 3  |-  ( Tr 
U. A  ->  ( U. A  C_  On  ->  ( Ord  On  ->  Ord  U. A ) ) )
2320, 22mpii 41 . 2  |-  ( Tr 
U. A  ->  ( U. A  C_  On  ->  Ord  U. A ) )
2413, 19, 23sylc 58 1  |-  ( A 
C_  On  ->  Ord  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   U.cuni 4007   Tr wtr 4294   Ord word 4572   Oncon0 4573
This theorem is referenced by:  ssonuni  4759  ssonprc  4764  orduni  4766  onsucuni  4800  limuni3  4824  onfununi  6595  tfrlem8  6637  onssnum  7913  unialeph  7974  cfslbn  8139  hsmexlem1  8298  inaprc  8703  nobndlem1  25639  nobndlem2  25640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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