MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1one2o Structured version   Visualization version   GIF version

Theorem 1one2o 8683
Description: Ordinal one is not ordinal two. Analogous to 1ne2 12472. (Contributed by AV, 17-Oct-2023.)
Assertion
Ref Expression
1one2o 1o ≠ 2o

Proof of Theorem 1one2o
StepHypRef Expression
1 1onn 8677 . . 3 1o ∈ ω
2 omsucne 7906 . . 3 (1o ∈ ω → 1o ≠ suc 1o)
31, 2ax-mp 5 . 2 1o ≠ suc 1o
4 df-2o 8506 . 2 2o = suc 1o
53, 4neeqtrri 3012 1 1o ≠ 2o
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wne 2938  suc csuc 6388  ωcom 7887  1oc1o 8498  2oc2o 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888  df-1o 8505  df-2o 8506
This theorem is referenced by:  gonanegoal  35337  satffunlem1lem1  35387  satffunlem2lem1  35389  ex-sategoelelomsuc  35411  ex-sategoelel12  35412
  Copyright terms: Public domain W3C validator