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Theorem ssorduni 4577
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 3831 . . . . 5  |-  ( x  e.  U. A  <->  E. y  e.  A  x  e.  y )
2 ssel 3174 . . . . . . . . 9  |-  ( A 
C_  On  ->  ( y  e.  A  ->  y  e.  On ) )
3 onelss 4434 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  C_  y ) )
42, 3syl6 29 . . . . . . . 8  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  C_  y ) ) )
5 anc2r 539 . . . . . . . 8  |-  ( ( y  e.  A  -> 
( x  e.  y  ->  x  C_  y
) )  ->  (
y  e.  A  -> 
( x  e.  y  ->  ( x  C_  y  /\  y  e.  A
) ) ) )
64, 5syl 15 . . . . . . 7  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  -> 
( x  C_  y  /\  y  e.  A
) ) ) )
7 ssuni 3849 . . . . . . 7  |-  ( ( x  C_  y  /\  y  e.  A )  ->  x  C_  U. A )
86, 7syl8 65 . . . . . 6  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  C_  U. A ) ) )
98rexlimdv 2666 . . . . 5  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  C_ 
U. A ) )
101, 9syl5bi 208 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  C_  U. A ) )
1110ralrimiv 2625 . . 3  |-  ( A 
C_  On  ->  A. x  e.  U. A x  C_  U. A )
12 dftr3 4117 . . 3  |-  ( Tr 
U. A  <->  A. x  e.  U. A x  C_  U. A )
1311, 12sylibr 203 . 2  |-  ( A 
C_  On  ->  Tr  U. A )
14 onelon 4417 . . . . . . 7  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
1514ex 423 . . . . . 6  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  e.  On )
)
162, 15syl6 29 . . . . 5  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  e.  On )
) )
1716rexlimdv 2666 . . . 4  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  e.  On ) )
181, 17syl5bi 208 . . 3  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  e.  On )
)
1918ssrdv 3185 . 2  |-  ( A 
C_  On  ->  U. A  C_  On )
20 ordon 4574 . . 3  |-  Ord  On
21 trssord 4409 . . . 4  |-  ( ( Tr  U. A  /\  U. A  C_  On  /\  Ord  On )  ->  Ord  U. A
)
22213exp 1150 . . 3  |-  ( Tr 
U. A  ->  ( U. A  C_  On  ->  ( Ord  On  ->  Ord  U. A ) ) )
2320, 22mpii 39 . 2  |-  ( Tr 
U. A  ->  ( U. A  C_  On  ->  Ord  U. A ) )
2413, 19, 23sylc 56 1  |-  ( A 
C_  On  ->  Ord  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827   Tr wtr 4113   Ord word 4391   Oncon0 4392
This theorem is referenced by:  ssonuni  4578  ssonprc  4583  orduni  4585  onsucuni  4619  limuni3  4643  onfununi  6358  tfrlem8  6400  onssnum  7667  unialeph  7728  cfslbn  7893  hsmexlem1  8052  inaprc  8458  nobndlem1  24346  nobndlem2  24347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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