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Theorem orduniss2 4624
Description: The union of the ordinal subsets of an ordinal number is that number. (Contributed by NM, 30-Jan-2005.)
Assertion
Ref Expression
orduniss2  |-  ( Ord 
A  ->  U. { x  e.  On  |  x  C_  A }  =  A
)
Distinct variable group:    x, A

Proof of Theorem orduniss2
StepHypRef Expression
1 df-rab 2552 . . . . 5  |-  { x  e.  On  |  x  C_  A }  =  {
x  |  ( x  e.  On  /\  x  C_  A ) }
2 incom 3361 . . . . . 6  |-  ( { x  |  x  e.  On }  i^i  {
x  |  x  C_  A } )  =  ( { x  |  x 
C_  A }  i^i  { x  |  x  e.  On } )
3 inab 3436 . . . . . 6  |-  ( { x  |  x  e.  On }  i^i  {
x  |  x  C_  A } )  =  {
x  |  ( x  e.  On  /\  x  C_  A ) }
4 df-pw 3627 . . . . . . . 8  |-  ~P A  =  { x  |  x 
C_  A }
54eqcomi 2287 . . . . . . 7  |-  { x  |  x  C_  A }  =  ~P A
6 abid2 2400 . . . . . . 7  |-  { x  |  x  e.  On }  =  On
75, 6ineq12i 3368 . . . . . 6  |-  ( { x  |  x  C_  A }  i^i  { x  |  x  e.  On } )  =  ( ~P A  i^i  On )
82, 3, 73eqtr3i 2311 . . . . 5  |-  { x  |  ( x  e.  On  /\  x  C_  A ) }  =  ( ~P A  i^i  On )
91, 8eqtri 2303 . . . 4  |-  { x  e.  On  |  x  C_  A }  =  ( ~P A  i^i  On )
10 ordpwsuc 4606 . . . 4  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )
119, 10syl5eq 2327 . . 3  |-  ( Ord 
A  ->  { x  e.  On  |  x  C_  A }  =  suc  A )
1211unieqd 3838 . 2  |-  ( Ord 
A  ->  U. { x  e.  On  |  x  C_  A }  =  U. suc  A )
13 ordunisuc 4623 . 2  |-  ( Ord 
A  ->  U. suc  A  =  A )
1412, 13eqtrd 2315 1  |-  ( Ord 
A  ->  U. { x  e.  On  |  x  C_  A }  =  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   Ord word 4391   Oncon0 4392   suc csuc 4394
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-pw 3627  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  df-suc 4398
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