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Theorem nlim0 4450
Description: The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
nlim0  |-  -.  Lim  (/)

Proof of Theorem nlim0
StepHypRef Expression
1 noel 3459 . . 3  |-  -.  (/)  e.  (/)
2 simp2 956 . . 3  |-  ( ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )  ->  (/) 
e.  (/) )
31, 2mto 167 . 2  |-  -.  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )
4 dflim2 4448 . 2  |-  ( Lim  (/) 
<->  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) ) )
53, 4mtbir 290 1  |-  -.  Lim  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ w3a 934    = wceq 1623    e. wcel 1684   (/)c0 3455   U.cuni 3827   Ord word 4391   Lim wlim 4393
This theorem is referenced by:  0ellim  4454  tz7.44lem1  6418  tz7.44-3  6421  cflim2  7889  rankcf  8399  dfrdg4  24488  limsucncmpi  24884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-lim 4397
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