Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sucneqoni Structured version   Visualization version   GIF version

Theorem sucneqoni 37334
Description: Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.)
Hypotheses
Ref Expression
sucneqoni.1 𝑋 = suc 𝑌
sucneqoni.2 𝑌 ∈ On
Assertion
Ref Expression
sucneqoni 𝑋𝑌

Proof of Theorem sucneqoni
StepHypRef Expression
1 sucneqoni.1 . . . 4 𝑋 = suc 𝑌
21a1i 11 . . 3 (⊤ → 𝑋 = suc 𝑌)
3 sucneqoni.2 . . . 4 𝑌 ∈ On
43a1i 11 . . 3 (⊤ → 𝑌 ∈ On)
52, 4sucneqond 37333 . 2 (⊤ → 𝑋𝑌)
65mptru 1544 1 𝑋𝑌
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wtru 1538  wcel 2108  wne 2946  Oncon0 6397  suc csuc 6399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6400  df-on 6401  df-suc 6403
This theorem is referenced by:  finxpreclem3  37361
  Copyright terms: Public domain W3C validator