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Theorem ordtoplem 36436
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 2941 . 2 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
2 ordeleqon 7802 . . . . . 6 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 unon 7851 . . . . . . . . 9 On = On
43eqcomi 2746 . . . . . . . 8 On = On
5 id 22 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
6 unieq 4918 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
74, 5, 63eqtr4a 2803 . . . . . . 7 (𝐴 = On → 𝐴 = 𝐴)
87orim2i 911 . . . . . 6 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
92, 8sylbi 217 . . . . 5 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
109orcomd 872 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 ∈ On))
1110ord 865 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
12 orduniorsuc 7850 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
1312ord 865 . . 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 2836 . . . 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 848   = wceq 1540  wcel 2108  wne 2940   cuni 4907  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  ordtop  36437  ordtopconn  36440  ordtopt0  36443
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