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Theorem orduniss2 4848
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 2721 . . . . 5  |-  { x  e.  On  |  x  C_  A }  =  {
x  |  ( x  e.  On  /\  x  C_  A ) }
2 incom 3522 . . . . . 6  |-  ( { x  |  x  e.  On }  i^i  {
x  |  x  C_  A } )  =  ( { x  |  x 
C_  A }  i^i  { x  |  x  e.  On } )
3 inab 3597 . . . . . 6  |-  ( { x  |  x  e.  On }  i^i  {
x  |  x  C_  A } )  =  {
x  |  ( x  e.  On  /\  x  C_  A ) }
4 df-pw 3830 . . . . . . . 8  |-  ~P A  =  { x  |  x 
C_  A }
54eqcomi 2447 . . . . . . 7  |-  { x  |  x  C_  A }  =  ~P A
6 abid2 2560 . . . . . . 7  |-  { x  |  x  e.  On }  =  On
75, 6ineq12i 3529 . . . . . 6  |-  ( { x  |  x  C_  A }  i^i  { x  |  x  e.  On } )  =  ( ~P A  i^i  On )
82, 3, 73eqtr3i 2471 . . . . 5  |-  { x  |  ( x  e.  On  /\  x  C_  A ) }  =  ( ~P A  i^i  On )
91, 8eqtri 2463 . . . 4  |-  { x  e.  On  |  x  C_  A }  =  ( ~P A  i^i  On )
10 ordpwsuc 4830 . . . 4  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )
119, 10syl5eq 2487 . . 3  |-  ( Ord 
A  ->  { x  e.  On  |  x  C_  A }  =  suc  A )
1211unieqd 4055 . 2  |-  ( Ord 
A  ->  U. { x  e.  On  |  x  C_  A }  =  U. suc  A )
13 ordunisuc 4847 . 2  |-  ( Ord 
A  ->  U. suc  A  =  A )
1412, 13eqtrd 2475 1  |-  ( Ord 
A  ->  U. { x  e.  On  |  x  C_  A }  =  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1654    e. wcel 1728   {cab 2429   {crab 2716    i^i cin 3308    C_ wss 3309   ~Pcpw 3828   U.cuni 4044   Ord word 4615   Oncon0 4616   suc csuc 4618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-tr 4334  df-eprel 4529  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-suc 4622
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