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

Theorem sucneqond 33711
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
sucneqond.1 (𝜑𝑋 = suc 𝑌)
sucneqond.2 (𝜑𝑌 ∈ On)
Assertion
Ref Expression
sucneqond (𝜑𝑋𝑌)

Proof of Theorem sucneqond
StepHypRef Expression
1 sucneqond.2 . . . . 5 (𝜑𝑌 ∈ On)
2 sucidg 6019 . . . . 5 (𝑌 ∈ On → 𝑌 ∈ suc 𝑌)
31, 2syl 17 . . . 4 (𝜑𝑌 ∈ suc 𝑌)
4 sucneqond.1 . . . 4 (𝜑𝑋 = suc 𝑌)
53, 4eleqtrrd 2881 . . 3 (𝜑𝑌𝑋)
6 suceloni 7247 . . . . . . . 8 (𝑌 ∈ On → suc 𝑌 ∈ On)
71, 6syl 17 . . . . . . 7 (𝜑 → suc 𝑌 ∈ On)
84, 7eqeltrd 2878 . . . . . 6 (𝜑𝑋 ∈ On)
9 eloni 5951 . . . . . 6 (𝑋 ∈ On → Ord 𝑋)
108, 9syl 17 . . . . 5 (𝜑 → Ord 𝑋)
11 ordirr 5959 . . . . 5 (Ord 𝑋 → ¬ 𝑋𝑋)
1210, 11syl 17 . . . 4 (𝜑 → ¬ 𝑋𝑋)
13 eleq1 2866 . . . . . 6 (𝑋 = 𝑌 → (𝑋𝑋𝑌𝑋))
1413biimprd 240 . . . . 5 (𝑋 = 𝑌 → (𝑌𝑋𝑋𝑋))
1514con3d 150 . . . 4 (𝑋 = 𝑌 → (¬ 𝑋𝑋 → ¬ 𝑌𝑋))
1612, 15syl5com 31 . . 3 (𝜑 → (𝑋 = 𝑌 → ¬ 𝑌𝑋))
175, 16mt2d 134 . 2 (𝜑 → ¬ 𝑋 = 𝑌)
1817neqned 2978 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1653  wcel 2157  wne 2971  Ord word 5940  Oncon0 5941  suc csuc 5943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945  df-suc 5947
This theorem is referenced by:  sucneqoni  33712
  Copyright terms: Public domain W3C validator