MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unizlim Unicode version

Theorem unizlim 4657
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 2569 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 df-lim 4546 . . . . . . . . 9  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
32biimpri 198 . . . . . . . 8  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  ->  Lim  A )
433exp 1152 . . . . . . 7  |-  ( Ord 
A  ->  ( A  =/=  (/)  ->  ( A  =  U. A  ->  Lim  A ) ) )
51, 4syl5bir 210 . . . . . 6  |-  ( Ord 
A  ->  ( -.  A  =  (/)  ->  ( A  =  U. A  ->  Lim  A ) ) )
65com23 74 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  ->  ( -.  A  =  (/)  ->  Lim  A ) ) )
76imp 419 . . . 4  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( -.  A  =  (/)  ->  Lim  A )
)
87orrd 368 . . 3  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( A  =  (/)  \/ 
Lim  A ) )
98ex 424 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  ->  ( A  =  (/)  \/  Lim  A ) ) )
10 uni0 4002 . . . . 5  |-  U. (/)  =  (/)
1110eqcomi 2408 . . . 4  |-  (/)  =  U. (/)
12 id 20 . . . 4  |-  ( A  =  (/)  ->  A  =  (/) )
13 unieq 3984 . . . 4  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
1411, 12, 133eqtr4a 2462 . . 3  |-  ( A  =  (/)  ->  A  = 
U. A )
15 limuni 4601 . . 3  |-  ( Lim 
A  ->  A  =  U. A )
1614, 15jaoi 369 . 2  |-  ( ( A  =  (/)  \/  Lim  A )  ->  A  =  U. A )
179, 16impbid1 195 1  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    =/= wne 2567   (/)c0 3588   U.cuni 3975   Ord word 4540   Lim wlim 4542
This theorem is referenced by:  ordzsl  4784  oeeulem  6803  cantnfp1lem2  7591  cantnflem1  7601  cnfcom2lem  7614  ordcmp  26101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-v 2918  df-dif 3283  df-in 3287  df-ss 3294  df-nul 3589  df-sn 3780  df-uni 3976  df-lim 4546
  Copyright terms: Public domain W3C validator