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Theorem dflim3 6997
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 5689 . 2 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
2 3anass 1040 . 2 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) ↔ (Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = 𝐴)))
3 df-ne 2791 . . . . . 6 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
43a1i 11 . . . . 5 (Ord 𝐴 → (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅))
5 orduninsuc 6993 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
64, 5anbi12d 746 . . . 4 (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
7 ioran 511 . . . 4 (¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (¬ 𝐴 = ∅ ∧ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
86, 7syl6bbr 278 . . 3 (Ord 𝐴 → ((𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) ↔ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
98pm5.32i 668 . 2 ((Ord 𝐴 ∧ (𝐴 ≠ ∅ ∧ 𝐴 = 𝐴)) ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
101, 2, 93bitri 286 1 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wne 2790  wrex 2908  c0 3893   cuni 4404  Ord word 5683  Oncon0 5684  Lim wlim 5685  suc csuc 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-tr 4715  df-eprel 4987  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690
This theorem is referenced by:  nlimon  7001  tfinds  7009  oalimcl  7588  omlimcl  7606  r1wunlim  9506
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