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Theorem ordtoplem 32966
 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 2999 . 2 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
2 ordeleqon 7248 . . . . . 6 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 unon 7291 . . . . . . . . 9 On = On
43eqcomi 2833 . . . . . . . 8 On = On
5 id 22 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
6 unieq 4665 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
74, 5, 63eqtr4a 2886 . . . . . . 7 (𝐴 = On → 𝐴 = 𝐴)
87orim2i 941 . . . . . 6 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
92, 8sylbi 209 . . . . 5 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
109orcomd 904 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 ∈ On))
1110ord 897 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
12 orduniorsuc 7290 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
1312ord 897 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
14 onuni 7253 . . . 4 (𝐴 ∈ On → 𝐴 ∈ On)
15 ordtoplem.1 . . . 4 ( 𝐴 ∈ On → suc 𝐴𝑆)
16 eleq1a 2900 . . . 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, 18syl5bi 234 1 (Ord 𝐴 → (𝐴 𝐴𝐴𝑆))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 880   = wceq 1658   ∈ wcel 2166   ≠ wne 2998  ∪ cuni 4657  Ord word 5961  Oncon0 5962  suc csuc 5964 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126  ax-un 7208 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-tr 4975  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-ord 5965  df-on 5966  df-suc 5968 This theorem is referenced by:  ordtop  32967  ordtopconn  32970  ordtopt0  32973
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