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Theorem unizlim 4994
Description: An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
Assertion
Ref Expression
unizlim  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )

Proof of Theorem unizlim
StepHypRef Expression
1 df-ne 2664 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 df-lim 4883 . . . . . . . . 9  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
32biimpri 206 . . . . . . . 8  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
433exp 1195 . . . . . . 7  |-  ( Ord 
A  ->  ( A  =/=  (/)  ->  ( A  =  U. A  ->  Lim  A ) ) )
51, 4syl5bir 218 . . . . . 6  |-  ( Ord 
A  ->  ( -.  A  =  (/)  ->  ( A  =  U. A  ->  Lim  A ) ) )
65com23 78 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  ->  ( -.  A  =  (/)  ->  Lim  A ) ) )
76imp 429 . . . 4  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( -.  A  =  (/)  ->  Lim  A )
)
87orrd 378 . . 3  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( A  =  (/)  \/ 
Lim  A ) )
98ex 434 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  ->  ( A  =  (/)  \/  Lim  A ) ) )
10 uni0 4272 . . . . 5  |-  U. (/)  =  (/)
1110eqcomi 2480 . . . 4  |-  (/)  =  U. (/)
12 id 22 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
13 unieq 4253 . . . 4  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
1411, 12, 133eqtr4a 2534 . . 3  |-  ( A  =  (/)  ->  A  = 
U. A )
15 limuni 4938 . . 3  |-  ( Lim 
A  ->  A  =  U. A )
1614, 15jaoi 379 . 2  |-  ( ( A  =  (/)  \/  Lim  A )  ->  A  =  U. A )
179, 16impbid1 203 1  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    =/= wne 2662   (/)c0 3785   U.cuni 4245   Ord word 4877   Lim wlim 4879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-uni 4246  df-lim 4883
This theorem is referenced by:  ordzsl  6665  oeeulem  7251  cantnfp1lem2  8099  cantnflem1  8109  cantnfp1lem2OLD  8125  cantnflem1OLD  8132  cnfcom2lem  8146  cnfcom2lemOLD  8154  ordcmp  29765
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