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Theorem sucneqond 33343
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 5841 . . . . 5 (𝑌 ∈ On → 𝑌 ∈ suc 𝑌)
31, 2syl 17 . . . 4 (𝜑𝑌 ∈ suc 𝑌)
4 sucneqond.1 . . . 4 (𝜑𝑋 = suc 𝑌)
53, 4eleqtrrd 2733 . . 3 (𝜑𝑌𝑋)
6 suceloni 7055 . . . . . . . 8 (𝑌 ∈ On → suc 𝑌 ∈ On)
71, 6syl 17 . . . . . . 7 (𝜑 → suc 𝑌 ∈ On)
84, 7eqeltrd 2730 . . . . . 6 (𝜑𝑋 ∈ On)
9 eloni 5771 . . . . . 6 (𝑋 ∈ On → Ord 𝑋)
108, 9syl 17 . . . . 5 (𝜑 → Ord 𝑋)
11 ordirr 5779 . . . . 5 (Ord 𝑋 → ¬ 𝑋𝑋)
1210, 11syl 17 . . . 4 (𝜑 → ¬ 𝑋𝑋)
13 eleq1 2718 . . . . . 6 (𝑋 = 𝑌 → (𝑋𝑋𝑌𝑋))
1413biimprd 238 . . . . 5 (𝑋 = 𝑌 → (𝑌𝑋𝑋𝑋))
1514con3d 148 . . . 4 (𝑋 = 𝑌 → (¬ 𝑋𝑋 → ¬ 𝑌𝑋))
1612, 15syl5com 31 . . 3 (𝜑 → (𝑋 = 𝑌 → ¬ 𝑌𝑋))
175, 16mt2d 131 . 2 (𝜑 → ¬ 𝑋 = 𝑌)
1817neqned 2830 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030  wne 2823  Ord word 5760  Oncon0 5761  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767
This theorem is referenced by:  sucneqoni  33344
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