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Theorem sucneqoni 34529
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 34528 . 2 (⊤ → 𝑋𝑌)
65mptru 1535 1 𝑋𝑌
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wtru 1529  wcel 2105  wne 3013  Oncon0 6184  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-suc 6190
This theorem is referenced by:  finxpreclem3  34556
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