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Theorem nlim0 4929
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 3782 . . 3  |-  -.  (/)  e.  (/)
2 simp2 992 . . 3  |-  ( ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )  ->  (/) 
e.  (/) )
31, 2mto 176 . 2  |-  -.  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )
4 dflim2 4927 . 2  |-  ( Lim  (/) 
<->  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) ) )
53, 4mtbir 299 1  |-  -.  Lim  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ w3a 968    = wceq 1374    e. wcel 1762   (/)c0 3778   U.cuni 4238   Ord word 4870   Lim wlim 4872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-lim 4876
This theorem is referenced by:  0ellim  4933  tz7.44lem1  7061  tz7.44-3  7064  cflim2  8632  rankcf  9144  dfrdg4  29027  limsucncmpi  29337
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