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Theorem dflim3 7836
Description: An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dflim3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem dflim3
StepHypRef Expression
1 df-lim 6370 . 2 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
2 3anass 1096 . 2 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = 𝐴)))
3 df-ne 2942 . . . . . 6 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
43a1i 11 . . . . 5 (Ord 𝐴 → (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅))
5 orduninsuc 7832 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
64, 5anbi12d 632 . . . 4 (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
7 ioran 983 . . . 4 (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
86, 7bitr4di 289 . . 3 (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) ↔ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
98pm5.32i 576 . 2 ((Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = 𝐴)) ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
101, 2, 93bitri 297 1 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wne 2941  wrex 3071  c0 4323   cuni 4909  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371
This theorem is referenced by:  nlimon  7840  tfinds  7849  oalimcl  8560  omlimcl  8578  r1wunlim  10732  dflim6  42014  naddgeoa  42145  faosnf0.11b  42178  dfsucon  42274
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