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

Theorem sucneqoni 33544
 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 33543 . 2 (⊤ → 𝑋𝑌)
65trud 1640 1 𝑋𝑌
 Colors of variables: wff setvar class Syntax hints:   = wceq 1630  ⊤wtru 1631   ∈ wcel 2144   ≠ wne 2942  Oncon0 5866  suc csuc 5868 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-tr 4885  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870  df-suc 5872 This theorem is referenced by:  finxpreclem3  33560
 Copyright terms: Public domain W3C validator