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 33550
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 5946 . . . . 5 (𝑌 ∈ On → 𝑌 ∈ suc 𝑌)
31, 2syl 17 . . . 4 (𝜑𝑌 ∈ suc 𝑌)
4 sucneqond.1 . . . 4 (𝜑𝑋 = suc 𝑌)
53, 4eleqtrrd 2853 . . 3 (𝜑𝑌𝑋)
6 suceloni 7160 . . . . . . . 8 (𝑌 ∈ On → suc 𝑌 ∈ On)
71, 6syl 17 . . . . . . 7 (𝜑 → suc 𝑌 ∈ On)
84, 7eqeltrd 2850 . . . . . 6 (𝜑𝑋 ∈ On)
9 eloni 5876 . . . . . 6 (𝑋 ∈ On → Ord 𝑋)
108, 9syl 17 . . . . 5 (𝜑 → Ord 𝑋)
11 ordirr 5884 . . . . 5 (Ord 𝑋 → ¬ 𝑋𝑋)
1210, 11syl 17 . . . 4 (𝜑 → ¬ 𝑋𝑋)
13 eleq1 2838 . . . . . 6 (𝑋 = 𝑌 → (𝑋𝑋𝑌𝑋))
1413biimprd 238 . . . . 5 (𝑋 = 𝑌 → (𝑌𝑋𝑋𝑋))
1514con3d 149 . . . 4 (𝑋 = 𝑌 → (¬ 𝑋𝑋 → ¬ 𝑌𝑋))
1612, 15syl5com 31 . . 3 (𝜑 → (𝑋 = 𝑌 → ¬ 𝑌𝑋))
175, 16mt2d 133 . 2 (𝜑 → ¬ 𝑋 = 𝑌)
1817neqned 2950 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1631  wcel 2145  wne 2943  Ord word 5865  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 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  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:  sucneqoni  33551
  Copyright terms: Public domain W3C validator