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Theorem orduniss2 4804
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 2706 . . . . 5  |-  { x  e.  On  |  x  C_  A }  =  {
x  |  ( x  e.  On  /\  x  C_  A ) }
2 incom 3525 . . . . . 6  |-  ( { x  |  x  e.  On }  i^i  {
x  |  x  C_  A } )  =  ( { x  |  x 
C_  A }  i^i  { x  |  x  e.  On } )
3 inab 3601 . . . . . 6  |-  ( { x  |  x  e.  On }  i^i  {
x  |  x  C_  A } )  =  {
x  |  ( x  e.  On  /\  x  C_  A ) }
4 df-pw 3793 . . . . . . . 8  |-  ~P A  =  { x  |  x 
C_  A }
54eqcomi 2439 . . . . . . 7  |-  { x  |  x  C_  A }  =  ~P A
6 abid2 2552 . . . . . . 7  |-  { x  |  x  e.  On }  =  On
75, 6ineq12i 3532 . . . . . 6  |-  ( { x  |  x  C_  A }  i^i  { x  |  x  e.  On } )  =  ( ~P A  i^i  On )
82, 3, 73eqtr3i 2463 . . . . 5  |-  { x  |  ( x  e.  On  /\  x  C_  A ) }  =  ( ~P A  i^i  On )
91, 8eqtri 2455 . . . 4  |-  { x  e.  On  |  x  C_  A }  =  ( ~P A  i^i  On )
10 ordpwsuc 4786 . . . 4  |-  ( Ord 
A  ->  ( ~P A  i^i  On )  =  suc  A )
119, 10syl5eq 2479 . . 3  |-  ( Ord 
A  ->  { x  e.  On  |  x  C_  A }  =  suc  A )
1211unieqd 4018 . 2  |-  ( Ord 
A  ->  U. { x  e.  On  |  x  C_  A }  =  U. suc  A )
13 ordunisuc 4803 . 2  |-  ( Ord 
A  ->  U. suc  A  =  A )
1412, 13eqtrd 2467 1  |-  ( Ord 
A  ->  U. { x  e.  On  |  x  C_  A }  =  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   {crab 2701    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   Ord word 4572   Oncon0 4573   suc csuc 4575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
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