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Theorem unizlim 4701
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 2603 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 df-lim 4589 . . . . . . . . 9  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
32biimpri 199 . . . . . . . 8  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
433exp 1153 . . . . . . 7  |-  ( Ord 
A  ->  ( A  =/=  (/)  ->  ( A  =  U. A  ->  Lim  A ) ) )
51, 4syl5bir 211 . . . . . 6  |-  ( Ord 
A  ->  ( -.  A  =  (/)  ->  ( A  =  U. A  ->  Lim  A ) ) )
65com23 75 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  ->  ( -.  A  =  (/)  ->  Lim  A ) ) )
76imp 420 . . . 4  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( -.  A  =  (/)  ->  Lim  A )
)
87orrd 369 . . 3  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( A  =  (/)  \/ 
Lim  A ) )
98ex 425 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  ->  ( A  =  (/)  \/  Lim  A ) ) )
10 uni0 4044 . . . . 5  |-  U. (/)  =  (/)
1110eqcomi 2442 . . . 4  |-  (/)  =  U. (/)
12 id 21 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
13 unieq 4026 . . . 4  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
1411, 12, 133eqtr4a 2496 . . 3  |-  ( A  =  (/)  ->  A  = 
U. A )
15 limuni 4644 . . 3  |-  ( Lim 
A  ->  A  =  U. A )
1614, 15jaoi 370 . 2  |-  ( ( A  =  (/)  \/  Lim  A )  ->  A  =  U. A )
179, 16impbid1 196 1  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    =/= wne 2601   (/)c0 3630   U.cuni 4017   Ord word 4583   Lim wlim 4585
This theorem is referenced by:  ordzsl  4828  oeeulem  6847  cantnfp1lem2  7638  cantnflem1  7648  cnfcom2lem  7661  ordcmp  26202
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 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-uni 4018  df-lim 4589
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