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Theorem onnbtwn 5816
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 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

Proof of Theorem onnbtwn
StepHypRef Expression
1 eloni 5731 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordnbtwn 5814 . 2 (Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
31, 2syl 17 1 (𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 1989  Ord word 5720  Oncon0 5721  suc csuc 5723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-tr 4751  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-ord 5724  df-on 5725  df-suc 5727
This theorem is referenced by:  ordunisuc2  7041  oalimcl  7637  omlimcl  7655  oneo  7658  nnneo  7728
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