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Theorem nlim0 4877
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 3741 . . 3  |-  -.  (/)  e.  (/)
2 simp2 989 . . 3  |-  ( ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )  ->  (/) 
e.  (/) )
31, 2mto 176 . 2  |-  -.  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )
4 dflim2 4875 . 2  |-  ( Lim  (/) 
<->  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) ) )
53, 4mtbir 299 1  |-  -.  Lim  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ w3a 965    = wceq 1370    e. wcel 1758   (/)c0 3737   U.cuni 4191   Ord word 4818   Lim wlim 4820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-tr 4486  df-eprel 4732  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-lim 4824
This theorem is referenced by:  0ellim  4881  tz7.44lem1  6963  tz7.44-3  6966  cflim2  8535  rankcf  9047  dfrdg4  28117  limsucncmpi  28427
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