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Theorem nlim0 4639
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 3632 . . 3  |-  -.  (/)  e.  (/)
2 simp2 958 . . 3  |-  ( ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )  ->  (/) 
e.  (/) )
31, 2mto 169 . 2  |-  -.  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )
4 dflim2 4637 . 2  |-  ( Lim  (/) 
<->  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) ) )
53, 4mtbir 291 1  |-  -.  Lim  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ w3a 936    = wceq 1652    e. wcel 1725   (/)c0 3628   U.cuni 4015   Ord word 4580   Lim wlim 4582
This theorem is referenced by:  0ellim  4643  tz7.44lem1  6663  tz7.44-3  6666  cflim2  8143  rankcf  8652  dfrdg4  25795  limsucncmpi  26195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-lim 4586
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