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Theorem ordtoplem 35976
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 2931 . 2 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
2 ordeleqon 7782 . . . . . 6 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 unon 7832 . . . . . . . . 9 On = On
43eqcomi 2734 . . . . . . . 8 On = On
5 id 22 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
6 unieq 4914 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
74, 5, 63eqtr4a 2791 . . . . . . 7 (𝐴 = On → 𝐴 = 𝐴)
87orim2i 908 . . . . . 6 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
92, 8sylbi 216 . . . . 5 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
109orcomd 869 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 ∈ On))
1110ord 862 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
12 orduniorsuc 7831 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
1312ord 862 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
14 onuni 7789 . . . 4 (𝐴 ∈ On → 𝐴 ∈ On)
15 ordtoplem.1 . . . 4 ( 𝐴 ∈ On → suc 𝐴𝑆)
16 eleq1a 2820 . . . 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 241 1 (Ord 𝐴 → (𝐴 𝐴𝐴𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 845   = wceq 1533  wcel 2098  wne 2930   cuni 4903  Ord word 6363  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 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368  df-suc 6370
This theorem is referenced by:  ordtop  35977  ordtopconn  35980  ordtopt0  35983
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