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

Theorem onnbtwn 5856
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 5771 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordnbtwn 5854 . 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 383  wcel 2030  Ord word 5760  Oncon0 5761  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767
This theorem is referenced by:  ordunisuc2  7086  oalimcl  7685  omlimcl  7703  oneo  7706  nnneo  7776
  Copyright terms: Public domain W3C validator