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Theorem orduniss2 4754
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 2659 . . . . 5  |-  { x  e.  On  |  x  C_  A }  =  {
x  |  ( x  e.  On  /\  x  C_  A ) }
2 incom 3477 . . . . . 6  |-  ( { x  |  x  e.  On }  i^i  {
x  |  x  C_  A } )  =  ( { x  |  x 
C_  A }  i^i  { x  |  x  e.  On } )
3 inab 3553 . . . . . 6  |-  ( { x  |  x  e.  On }  i^i  {
x  |  x  C_  A } )  =  {
x  |  ( x  e.  On  /\  x  C_  A ) }
4 df-pw 3745 . . . . . . . 8  |-  ~P A  =  { x  |  x 
C_  A }
54eqcomi 2392 . . . . . . 7  |-  { x  |  x  C_  A }  =  ~P A
6 abid2 2505 . . . . . . 7  |-  { x  |  x  e.  On }  =  On
75, 6ineq12i 3484 . . . . . 6  |-  ( { x  |  x  C_  A }  i^i  { x  |  x  e.  On } )  =  ( ~P A  i^i  On )
82, 3, 73eqtr3i 2416 . . . . 5  |-  { x  |  ( x  e.  On  /\  x  C_  A ) }  =  ( ~P A  i^i  On )
91, 8eqtri 2408 . . . 4  |-  { x  e.  On  |  x  C_  A }  =  ( ~P A  i^i  On )
10 ordpwsuc 4736 . . . 4  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )
119, 10syl5eq 2432 . . 3  |-  ( Ord 
A  ->  { x  e.  On  |  x  C_  A }  =  suc  A )
1211unieqd 3969 . 2  |-  ( Ord 
A  ->  U. { x  e.  On  |  x  C_  A }  =  U. suc  A )
13 ordunisuc 4753 . 2  |-  ( Ord 
A  ->  U. suc  A  =  A )
1412, 13eqtrd 2420 1  |-  ( Ord 
A  ->  U. { x  e.  On  |  x  C_  A }  =  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2374   {crab 2654    i^i cin 3263    C_ wss 3264   ~Pcpw 3743   U.cuni 3958   Ord word 4522   Oncon0 4523   suc csuc 4525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529
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