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Theorem sucneqoni 36711
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 36710 . 2 (⊤ → 𝑋𝑌)
65mptru 1547 1 𝑋𝑌
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wtru 1541  wcel 2105  wne 2939  Oncon0 6364  suc csuc 6366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-suc 6370
This theorem is referenced by:  finxpreclem3  36738
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