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Theorem onnbtwn 4975
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 4894 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordnbtwn 4974 . 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 1767   Ord word 4883   Oncon0 4884   suc csuc 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890
This theorem is referenced by:  ordunisuc2  6674  oalimcl  7221  omlimcl  7239  oneo  7242  nnneo  7312
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