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Theorem unizlim 4588
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 2523 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 df-lim 4476 . . . . . . . . 9  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
32biimpri 197 . . . . . . . 8  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
433exp 1150 . . . . . . 7  |-  ( Ord 
A  ->  ( A  =/=  (/)  ->  ( A  =  U. A  ->  Lim  A ) ) )
51, 4syl5bir 209 . . . . . 6  |-  ( Ord 
A  ->  ( -.  A  =  (/)  ->  ( A  =  U. A  ->  Lim  A ) ) )
65com23 72 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  ->  ( -.  A  =  (/)  ->  Lim  A ) ) )
76imp 418 . . . 4  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( -.  A  =  (/)  ->  Lim  A )
)
87orrd 367 . . 3  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( A  =  (/)  \/ 
Lim  A ) )
98ex 423 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  ->  ( A  =  (/)  \/  Lim  A ) ) )
10 uni0 3933 . . . . 5  |-  U. (/)  =  (/)
1110eqcomi 2362 . . . 4  |-  (/)  =  U. (/)
12 id 19 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
13 unieq 3915 . . . 4  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
1411, 12, 133eqtr4a 2416 . . 3  |-  ( A  =  (/)  ->  A  = 
U. A )
15 limuni 4531 . . 3  |-  ( Lim 
A  ->  A  =  U. A )
1614, 15jaoi 368 . 2  |-  ( ( A  =  (/)  \/  Lim  A )  ->  A  =  U. A )
179, 16impbid1 194 1  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1642    =/= wne 2521   (/)c0 3531   U.cuni 3906   Ord word 4470   Lim wlim 4472
This theorem is referenced by:  ordzsl  4715  oeeulem  6683  cantnfp1lem2  7468  cantnflem1  7478  cnfcom2lem  7491  ordcmp  25445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-v 2866  df-dif 3231  df-in 3235  df-ss 3242  df-nul 3532  df-sn 3722  df-uni 3907  df-lim 4476
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