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Theorem ordtoplem 36418
Description: Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
Hypothesis
Ref Expression
ordtoplem.1 ( 𝐴 ∈ On → suc 𝐴𝑆)
Assertion
Ref Expression
ordtoplem (Ord 𝐴 → (𝐴 𝐴𝐴𝑆))

Proof of Theorem ordtoplem
StepHypRef Expression
1 df-ne 2939 . 2 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
2 ordeleqon 7801 . . . . . 6 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 unon 7851 . . . . . . . . 9 On = On
43eqcomi 2744 . . . . . . . 8 On = On
5 id 22 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
6 unieq 4923 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
74, 5, 63eqtr4a 2801 . . . . . . 7 (𝐴 = On → 𝐴 = 𝐴)
87orim2i 910 . . . . . 6 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
92, 8sylbi 217 . . . . 5 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
109orcomd 871 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 ∈ On))
1110ord 864 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
12 orduniorsuc 7850 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
1312ord 864 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
14 onuni 7808 . . . 4 (𝐴 ∈ On → 𝐴 ∈ On)
15 ordtoplem.1 . . . 4 ( 𝐴 ∈ On → suc 𝐴𝑆)
16 eleq1a 2834 . . . 4 (suc 𝐴𝑆 → (𝐴 = suc 𝐴𝐴𝑆))
1714, 15, 163syl 18 . . 3 (𝐴 ∈ On → (𝐴 = suc 𝐴𝐴𝑆))
1811, 13, 17syl6c 70 . 2 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴𝑆))
191, 18biimtrid 242 1 (Ord 𝐴 → (𝐴 𝐴𝐴𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 847   = wceq 1537  wcel 2106  wne 2938   cuni 4912  Ord word 6385  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  ordtop  36419  ordtopconn  36422  ordtopt0  36425
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