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Theorem onsucuni3 37368
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 6394 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
213ad2ant1 1134 . . . 4 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵)
3 orduniorsuc 7850 . . . 4 (Ord 𝐵 → (𝐵 = 𝐵𝐵 = suc 𝐵))
42, 3syl 17 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = 𝐵𝐵 = suc 𝐵))
54orcomd 872 . 2 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = suc 𝐵𝐵 = 𝐵))
6 simp2 1138 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅)
7 df-lim 6389 . . . . . . . 8 (Lim 𝐵 ↔ (Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
87biimpri 228 . . . . . . 7 ((Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵) → Lim 𝐵)
983expb 1121 . . . . . 6 ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)) → Lim 𝐵)
109con3i 154 . . . . 5 (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
11103ad2ant3 1136 . . . 4 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
122, 11mpnanrd 409 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
136, 12mpnanrd 409 . 2 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ 𝐵 = 𝐵)
14 orcom 871 . . 3 ((𝐵 = suc 𝐵𝐵 = 𝐵) ↔ (𝐵 = 𝐵𝐵 = suc 𝐵))
15 df-or 849 . . 3 ((𝐵 = 𝐵𝐵 = suc 𝐵) ↔ (¬ 𝐵 = 𝐵𝐵 = suc 𝐵))
1614, 15sylbb 219 . 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 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  c0 4333   cuni 4907  Ord word 6383  Oncon0 6384  Lim wlim 6385  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
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-lim 6389  df-suc 6390
This theorem is referenced by:  1oequni2o  37369  rdgsucuni  37370  finxpreclem4  37395
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