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Theorem onnbtwn 4483
Description: There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 9-Jun-1994.)
Assertion
Ref Expression
onnbtwn  |-  ( A  e.  On  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )

Proof of Theorem onnbtwn
StepHypRef Expression
1 eloni 4401 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordnbtwn 4482 . 2  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
31, 2syl 15 1  |-  ( A  e.  On  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1685   Ord word 4390   Oncon0 4391   suc csuc 4393
This theorem is referenced by:  ordunisuc2  4634  oalimcl  6554  omlimcl  6572  oneo  6575  nnneo  6645
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 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397
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