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Theorem ssorduni 6594
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 4239 . . . . 5  |-  ( x  e.  U. A  <->  E. y  e.  A  x  e.  y )
2 ssel 3483 . . . . . . . . 9  |-  ( A 
C_  On  ->  ( y  e.  A  ->  y  e.  On ) )
3 onelss 4909 . . . . . . . . 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 554 . . . . . . . 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 4257 . . . . . . 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 2944 . . . . 5  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  C_ 
U. A ) )
101, 9syl5bi 217 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  C_  U. A ) )
1110ralrimiv 2866 . . 3  |-  ( A 
C_  On  ->  A. x  e.  U. A x  C_  U. A )
12 dftr3 4536 . . 3  |-  ( Tr 
U. A  <->  A. x  e.  U. A x  C_  U. A )
1311, 12sylibr 212 . 2  |-  ( A 
C_  On  ->  Tr  U. A )
14 onelon 4892 . . . . . . 7  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
1514ex 432 . . . . . 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 2944 . . . 4  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  e.  On ) )
181, 17syl5bi 217 . . 3  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  e.  On )
)
1918ssrdv 3495 . 2  |-  ( A 
C_  On  ->  U. A  C_  On )
20 ordon 6591 . . 3  |-  Ord  On
21 trssord 4884 . . . 4  |-  ( ( Tr  U. A  /\  U. A  C_  On  /\  Ord  On )  ->  Ord  U. A
)
22213exp 1193 . . 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 60 1  |-  ( A 
C_  On  ->  Ord  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461   U.cuni 4235   Tr wtr 4532   Ord word 4866   Oncon0 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871
This theorem is referenced by:  ssonuni  6595  ssonprc  6600  orduni  6602  onsucuni  6636  limuni3  6660  onfununi  7004  tfrlem8  7045  onssnum  8412  unialeph  8473  cfslbn  8638  hsmexlem1  8797  inaprc  9203  nobndlem1  29692  nobndlem2  29693
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