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Theorem sucneqond 33193
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 5801 . . . . 5 (𝑌 ∈ On → 𝑌 ∈ suc 𝑌)
31, 2syl 17 . . . 4 (𝜑𝑌 ∈ suc 𝑌)
4 sucneqond.1 . . . 4 (𝜑𝑋 = suc 𝑌)
53, 4eleqtrrd 2703 . . 3 (𝜑𝑌𝑋)
6 suceloni 7010 . . . . . . . 8 (𝑌 ∈ On → suc 𝑌 ∈ On)
71, 6syl 17 . . . . . . 7 (𝜑 → suc 𝑌 ∈ On)
84, 7eqeltrd 2700 . . . . . 6 (𝜑𝑋 ∈ On)
9 eloni 5731 . . . . . 6 (𝑋 ∈ On → Ord 𝑋)
108, 9syl 17 . . . . 5 (𝜑 → Ord 𝑋)
11 ordirr 5739 . . . . 5 (Ord 𝑋 → ¬ 𝑋𝑋)
1210, 11syl 17 . . . 4 (𝜑 → ¬ 𝑋𝑋)
13 eleq1 2688 . . . . . 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 2800 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1482  wcel 1989  wne 2793  Ord word 5720  Oncon0 5721  suc csuc 5723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-tr 4751  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-ord 5724  df-on 5725  df-suc 5727
This theorem is referenced by:  sucneqoni  33194
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