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Theorem dflim4 6619
Description: An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)
Assertion
Ref Expression
dflim4  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
Distinct variable group:    x, A

Proof of Theorem dflim4
StepHypRef Expression
1 dflim2 4875 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
2 ordunisuc2 6615 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
32anbi2d 702 . . . 4  |-  ( Ord 
A  ->  ( ( (/) 
e.  A  /\  A  =  U. A )  <->  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A ) ) )
43pm5.32i 635 . . 3  |-  ( ( Ord  A  /\  ( (/) 
e.  A  /\  A  =  U. A ) )  <-> 
( Ord  A  /\  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
) )
5 3anass 976 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A )  <->  ( Ord  A  /\  ( (/)  e.  A  /\  A  =  U. A ) ) )
6 3anass 976 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  <-> 
( Ord  A  /\  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
) )
74, 5, 63bitr4i 277 . 2  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A )  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
81, 7bitri 249 1  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   A.wral 2751   (/)c0 3735   U.cuni 4188   Ord word 4818   Lim wlim 4820   suc csuc 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-tr 4487  df-eprel 4731  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825
This theorem is referenced by:  limsuc  6620  limuni3  6623  oelimcl  7204
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