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Theorem ordtoplem 33843
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 3015 . 2 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
2 ordeleqon 7497 . . . . . 6 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
3 unon 7540 . . . . . . . . 9 On = On
43eqcomi 2833 . . . . . . . 8 On = On
5 id 22 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
6 unieq 4835 . . . . . . . 8 (𝐴 = On → 𝐴 = On)
74, 5, 63eqtr4a 2885 . . . . . . 7 (𝐴 = On → 𝐴 = 𝐴)
87orim2i 908 . . . . . 6 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
92, 8sylbi 220 . . . . 5 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = 𝐴))
109orcomd 868 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 ∈ On))
1110ord 861 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 ∈ On))
12 orduniorsuc 7539 . . . 4 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
1312ord 861 . . 3 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
14 onuni 7502 . . . 4 (𝐴 ∈ On → 𝐴 ∈ On)
15 ordtoplem.1 . . . 4 ( 𝐴 ∈ On → suc 𝐴𝑆)
16 eleq1a 2911 . . . 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 245 1 (Ord 𝐴 → (𝐴 𝐴𝐴𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 844   = wceq 1538  wcel 2115  wne 3014   cuni 4824  Ord word 6177  Oncon0 6178  suc csuc 6180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182  df-suc 6184
This theorem is referenced by:  ordtop  33844  ordtopconn  33847  ordtopt0  33850
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