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Theorem onnbtwn 5716
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 5631 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordnbtwn 5714 . 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 382  wcel 1975  Ord word 5620  Oncon0 5621  suc csuc 5623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-tr 4670  df-eprel 4934  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-ord 5624  df-on 5625  df-suc 5627
This theorem is referenced by:  ordunisuc2  6908  oalimcl  7499  omlimcl  7517  oneo  7520  nnneo  7590
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