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Theorem dflim6 43254
Description: A limit ordinal is a non-zero ordinal which is not a successor ordinal. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
Assertion
Ref Expression
dflim6 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
Distinct variable group:   𝐴,𝑏

Proof of Theorem dflim6
StepHypRef Expression
1 ioran 985 . . . 4 (¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
2 df-ne 2939 . . . . 5 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
32anbi1i 624 . . . 4 ((𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
41, 3bitr4i 278 . . 3 (¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
54anbi2i 623 . 2 ((Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏)) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
6 dflim3 7868 . 2 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
7 3anass 1094 . 2 ((Ord 𝐴𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)))
85, 6, 73bitr4i 303 1 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wne 2938  wrex 3068  c0 4339  Ord word 6385  Oncon0 6386  Lim wlim 6387  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392
This theorem is referenced by:  limnsuc  43255
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