MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onnbtwn Structured version   Unicode version

Theorem onnbtwn 4959
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 4878 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordnbtwn 4958 . 2  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
31, 2syl 16 1  |-  ( A  e.  On  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1804   Ord word 4867   Oncon0 4868   suc csuc 4870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-suc 4874
This theorem is referenced by:  ordunisuc2  6664  oalimcl  7211  omlimcl  7229  oneo  7232  nnneo  7302
  Copyright terms: Public domain W3C validator