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Theorem onsucuni3 33497
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 5882 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
213ad2ant1 1125 . . . 4 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵)
3 orduniorsuc 7183 . . . 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 1129 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅)
7 df-lim 5877 . . . . . . . 8 (Lim 𝐵 ↔ (Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
87biimpri 218 . . . . . . 7 ((Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵) → Lim 𝐵)
983expb 1113 . . . . . 6 ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)) → Lim 𝐵)
109con3i 150 . . . . 5 (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
11103ad2ant3 1127 . . . 4 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
122, 11mpnanrd 33460 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → ¬ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
136, 12mpnanrd 33460 . 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 209 . 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  wo 382  wa 383  w3a 1072   = wceq 1620  wcel 2127  wne 2920  c0 4046   cuni 4576  Ord word 5871  Oncon0 5872  Lim wlim 5873  suc csuc 5874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-tr 4893  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878
This theorem is referenced by:  1oequni2o  33498  rdgsucuni  33499  finxpreclem4  33513
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