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Theorem onsucuni3 32391
 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 5650 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
213ad2ant1 1075 . . . 4 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵)
3 orduniorsuc 6922 . . . 4 (Ord 𝐵 → (𝐵 = 𝐵𝐵 = suc 𝐵))
42, 3syl 17 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = 𝐵𝐵 = suc 𝐵))
54orcomd 402 . 2 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = suc 𝐵𝐵 = 𝐵))
6 simp2 1055 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅)
7 df-lim 5645 . . . . . . . 8 (Lim 𝐵 ↔ (Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
87biimpri 217 . . . . . . 7 ((Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵) → Lim 𝐵)
983expb 1258 . . . . . 6 ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)) → Lim 𝐵)
109con3i 149 . . . . 5 (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
11103ad2ant3 1077 . . . 4 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
122, 11mpnanrd 32354 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
136, 12mpnanrd 32354 . 2 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ 𝐵 = 𝐵)
14 orcom 401 . . 3 ((𝐵 = suc 𝐵𝐵 = 𝐵) ↔ (𝐵 = 𝐵𝐵 = suc 𝐵))
15 df-or 384 . . 3 ((𝐵 = 𝐵𝐵 = suc 𝐵) ↔ (¬ 𝐵 = 𝐵𝐵 = suc 𝐵))
1614, 15sylbb 208 . 2 ((𝐵 = suc 𝐵𝐵 = 𝐵) → (¬ 𝐵 = 𝐵𝐵 = suc 𝐵))
175, 13, 16sylc 63 1 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ∪ cuni 4372  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646 This theorem is referenced by:  1oequni2o  32392  rdgsucuni  32393  finxpreclem4  32407
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