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Theorem ssorduni 6624
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 4221 . . . . 5  |-  ( x  e.  U. A  <->  E. y  e.  A  x  e.  y )
2 ssel 3459 . . . . . . . . 9  |-  ( A 
C_  On  ->  ( y  e.  A  ->  y  e.  On ) )
3 onelss 5482 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  C_  y ) )
42, 3syl6 35 . . . . . . . 8  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  C_  y ) ) )
5 anc2r 559 . . . . . . . 8  |-  ( ( y  e.  A  -> 
( x  e.  y  ->  x  C_  y
) )  ->  (
y  e.  A  -> 
( x  e.  y  ->  ( x  C_  y  /\  y  e.  A
) ) ) )
64, 5syl 17 . . . . . . 7  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  -> 
( x  C_  y  /\  y  e.  A
) ) ) )
7 ssuni 4239 . . . . . . 7  |-  ( ( x  C_  y  /\  y  e.  A )  ->  x  C_  U. A )
86, 7syl8 73 . . . . . 6  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  C_  U. A ) ) )
98rexlimdv 2916 . . . . 5  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  C_ 
U. A ) )
101, 9syl5bi 221 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  C_  U. A ) )
1110ralrimiv 2838 . . 3  |-  ( A 
C_  On  ->  A. x  e.  U. A x  C_  U. A )
12 dftr3 4520 . . 3  |-  ( Tr 
U. A  <->  A. x  e.  U. A x  C_  U. A )
1311, 12sylibr 216 . 2  |-  ( A 
C_  On  ->  Tr  U. A )
14 onelon 5465 . . . . . . 7  |-  ( ( y  e.  On  /\  x  e.  y )  ->  x  e.  On )
1514ex 436 . . . . . 6  |-  ( y  e.  On  ->  (
x  e.  y  ->  x  e.  On )
)
162, 15syl6 35 . . . . 5  |-  ( A 
C_  On  ->  ( y  e.  A  ->  (
x  e.  y  ->  x  e.  On )
) )
1716rexlimdv 2916 . . . 4  |-  ( A 
C_  On  ->  ( E. y  e.  A  x  e.  y  ->  x  e.  On ) )
181, 17syl5bi 221 . . 3  |-  ( A 
C_  On  ->  ( x  e.  U. A  ->  x  e.  On )
)
1918ssrdv 3471 . 2  |-  ( A 
C_  On  ->  U. A  C_  On )
20 ordon 6621 . . 3  |-  Ord  On
21 trssord 5457 . . . 4  |-  ( ( Tr  U. A  /\  U. A  C_  On  /\  Ord  On )  ->  Ord  U. A
)
22213exp 1205 . . 3  |-  ( Tr 
U. A  ->  ( U. A  C_  On  ->  ( Ord  On  ->  Ord  U. A ) ) )
2320, 22mpii 45 . 2  |-  ( Tr 
U. A  ->  ( U. A  C_  On  ->  Ord  U. A ) )
2413, 19, 23sylc 63 1  |-  ( A 
C_  On  ->  Ord  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1869   A.wral 2776   E.wrex 2777    C_ wss 3437   U.cuni 4217   Tr wtr 4516   Ord word 5439   Oncon0 5440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-tr 4517  df-eprel 4762  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-ord 5443  df-on 5444
This theorem is referenced by:  ssonuni  6625  ssonprc  6631  orduni  6633  onsucuni  6667  limuni3  6691  onfununi  7066  tfrlem8  7108  onssnum  8473  unialeph  8534  cfslbn  8699  hsmexlem1  8858  inaprc  9263  nobndlem1  30580  nobndlem2  30581
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