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Theorem sucneqond 35774
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 6396 . . . . 5 (𝑌 ∈ On → 𝑌 ∈ suc 𝑌)
31, 2syl 17 . . . 4 (𝜑𝑌 ∈ suc 𝑌)
4 sucneqond.1 . . . 4 (𝜑𝑋 = suc 𝑌)
53, 4eleqtrrd 2841 . . 3 (𝜑𝑌𝑋)
6 onsuc 7738 . . . . . . . 8 (𝑌 ∈ On → suc 𝑌 ∈ On)
71, 6syl 17 . . . . . . 7 (𝜑 → suc 𝑌 ∈ On)
84, 7eqeltrd 2838 . . . . . 6 (𝜑𝑋 ∈ On)
9 eloni 6325 . . . . . 6 (𝑋 ∈ On → Ord 𝑋)
108, 9syl 17 . . . . 5 (𝜑 → Ord 𝑋)
11 ordirr 6333 . . . . 5 (Ord 𝑋 → ¬ 𝑋𝑋)
1210, 11syl 17 . . . 4 (𝜑 → ¬ 𝑋𝑋)
13 eleq1 2825 . . . . . 6 (𝑋 = 𝑌 → (𝑋𝑋𝑌𝑋))
1413biimprd 247 . . . . 5 (𝑋 = 𝑌 → (𝑌𝑋𝑋𝑋))
1514con3d 152 . . . 4 (𝑋 = 𝑌 → (¬ 𝑋𝑋 → ¬ 𝑌𝑋))
1612, 15syl5com 31 . . 3 (𝜑 → (𝑋 = 𝑌 → ¬ 𝑌𝑋))
175, 16mt2d 136 . 2 (𝜑 → ¬ 𝑋 = 𝑌)
1817neqned 2948 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1541  wcel 2106  wne 2941  Ord word 6314  Oncon0 6315  suc csuc 6317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-tr 5221  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-ord 6318  df-on 6319  df-suc 6321
This theorem is referenced by:  sucneqoni  35775
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