Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onsucuni3 Structured version   Visualization version   GIF version

Theorem onsucuni3 37349
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 6395 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
213ad2ant1 1132 . . . 4 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → Ord 𝐵)
3 orduniorsuc 7849 . . . 4 (Ord 𝐵 → (𝐵 = 𝐵𝐵 = suc 𝐵))
42, 3syl 17 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = 𝐵𝐵 = suc 𝐵))
54orcomd 871 . 2 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (𝐵 = suc 𝐵𝐵 = 𝐵))
6 simp2 1136 . . 3 ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 ≠ ∅)
7 df-lim 6390 . . . . . . . 8 (Lim 𝐵 ↔ (Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵))
87biimpri 228 . . . . . . 7 ((Ord 𝐵𝐵 ≠ ∅ ∧ 𝐵 = 𝐵) → Lim 𝐵)
983expb 1119 . . . . . 6 ((Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)) → Lim 𝐵)
109con3i 154 . . . . 5 (¬ Lim 𝐵 → ¬ (Ord 𝐵 ∧ (𝐵 ≠ ∅ ∧ 𝐵 = 𝐵)))
11103ad2ant3 1134 . . . 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 870 . . 3 ((𝐵 = suc 𝐵𝐵 = 𝐵) ↔ (𝐵 = 𝐵𝐵 = suc 𝐵))
15 df-or 848 . . 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 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  c0 4338   cuni 4911  Ord word 6384  Oncon0 6385  Lim wlim 6386  suc csuc 6387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391
This theorem is referenced by:  1oequni2o  37350  rdgsucuni  37351  finxpreclem4  37376
  Copyright terms: Public domain W3C validator