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Theorem onsucuni3 37074
Description: If an ordinal number has a predecessor, then it is successor of that predecessor. (Contributed by ML, 17-Oct-2020.)
Assertion
Ref Expression
onsucuni3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc 𝐵)

Proof of Theorem onsucuni3
StepHypRef Expression
1 eloni 6386 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
213ad2ant1 1130 . . . 4 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵)
3 orduniorsuc 7839 . . . 4 (Ord 𝐵 → (𝐵 = 𝐵𝐵 = suc 𝐵))
42, 3syl 17 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = 𝐵𝐵 = suc 𝐵))
54orcomd 869 . 2 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = suc 𝐵𝐵 = 𝐵))
6 simp2 1134 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅)
7 df-lim 6381 . . . . . . . 8 (Lim 𝐵 ↔ (Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
87biimpri 227 . . . . . . 7 ((Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵) → Lim 𝐵)
983expb 1117 . . . . . 6 ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)) → Lim 𝐵)
109con3i 154 . . . . 5 (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
11103ad2ant3 1132 . . . 4 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
122, 11mpnanrd 408 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
136, 12mpnanrd 408 . 2 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ 𝐵 = 𝐵)
14 orcom 868 . . 3 ((𝐵 = suc 𝐵𝐵 = 𝐵) ↔ (𝐵 = 𝐵𝐵 = suc 𝐵))
15 df-or 846 . . 3 ((𝐵 = 𝐵𝐵 = suc 𝐵) ↔ (¬ 𝐵 = 𝐵𝐵 = suc 𝐵))
1614, 15sylbb 218 . 2 ((𝐵 = suc 𝐵𝐵 = 𝐵) → (¬ 𝐵 = 𝐵𝐵 = suc 𝐵))
175, 13, 16sylc 65 1 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  wne 2930  c0 4325   cuni 4913  Ord word 6375  Oncon0 6376  Lim wlim 6377  suc csuc 6378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382
This theorem is referenced by:  1oequni2o  37075  rdgsucuni  37076  finxpreclem4  37101
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