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Theorem unizlim 4983
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 2651 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 df-lim 4872 . . . . . . . . 9  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
32biimpri 206 . . . . . . . 8  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
433exp 1193 . . . . . . 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 427 . . . 4  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( -.  A  =  (/)  ->  Lim  A )
)
87orrd 376 . . 3  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( A  =  (/)  \/ 
Lim  A ) )
98ex 432 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  ->  ( A  =  (/)  \/  Lim  A ) ) )
10 uni0 4262 . . . . 5  |-  U. (/)  =  (/)
1110eqcomi 2467 . . . 4  |-  (/)  =  U. (/)
12 id 22 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
13 unieq 4243 . . . 4  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
1411, 12, 133eqtr4a 2521 . . 3  |-  ( A  =  (/)  ->  A  = 
U. A )
15 limuni 4927 . . 3  |-  ( Lim 
A  ->  A  =  U. A )
1614, 15jaoi 377 . 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    =/= wne 2649   (/)c0 3783   U.cuni 4235   Ord word 4866   Lim wlim 4868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-uni 4236  df-lim 4872
This theorem is referenced by:  ordzsl  6653  oeeulem  7242  cantnfp1lem2  8089  cantnflem1  8099  cantnfp1lem2OLD  8115  cantnflem1OLD  8122  cnfcom2lem  8136  cnfcom2lemOLD  8144  ordcmp  30140
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