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Theorem sucneqoni 37281
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 37280 . 2 (⊤ → 𝑋𝑌)
65mptru 1544 1 𝑋𝑌
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wtru 1538  wcel 2103  wne 2942  Oncon0 6394  suc csuc 6396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-tr 5287  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-ord 6397  df-on 6398  df-suc 6400
This theorem is referenced by:  finxpreclem3  37308
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