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Theorem dflim3 7788
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 6327 . 2 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
2 3anass 1096 . 2 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = 𝐴)))
3 df-ne 2945 . . . . . 6 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
43a1i 11 . . . . 5 (Ord 𝐴 → (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅))
5 orduninsuc 7784 . . . . 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 2944  wrex 3074  c0 4287   cuni 4870  Ord word 6321  Oncon0 6322  Lim wlim 6323  suc csuc 6324
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328
This theorem is referenced by:  nlimon  7792  tfinds  7801  oalimcl  8512  omlimcl  8530  r1wunlim  10680  dflim6  41628  naddgeoa  41740  faosnf0.11b  41773  dfsucon  41869
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