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Theorem ordzsl 6992
Description: An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
Assertion
Ref Expression
ordzsl (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ordzsl
StepHypRef Expression
1 orduninsuc 6990 . . . . . 6 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
21biimprd 238 . . . . 5 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥𝐴 = 𝐴))
3 unizlim 5803 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
42, 3sylibd 229 . . . 4 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴)))
54orrd 393 . . 3 (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
6 3orass 1039 . . . 4 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)))
7 or12 545 . . . 4 ((𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
86, 7bitri 264 . . 3 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
95, 8sylibr 224 . 2 (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
10 ord0 5736 . . . 4 Ord ∅
11 ordeq 5689 . . . 4 (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅))
1210, 11mpbiri 248 . . 3 (𝐴 = ∅ → Ord 𝐴)
13 suceloni 6960 . . . . . 6 (𝑥 ∈ On → suc 𝑥 ∈ On)
14 eleq1 2686 . . . . . 6 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1513, 14syl5ibr 236 . . . . 5 (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On))
16 eloni 5692 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
1715, 16syl6com 37 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴))
1817rexlimiv 3020 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴)
19 limord 5743 . . 3 (Lim 𝐴 → Ord 𝐴)
2012, 18, 193jaoi 1388 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) → Ord 𝐴)
219, 20impbii 199 1 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383  w3o 1035   = wceq 1480  wcel 1987  wrex 2908  c0 3891   cuni 4402  Ord word 5681  Oncon0 5682  Lim wlim 5683  suc csuc 5684
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 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
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 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688
This theorem is referenced by:  onzsl  6993  tfrlem16  7434  omeulem1  7607  oaabs2  7670  rankxplim3  8688  rankxpsuc  8689  cardlim  8742  cardaleph  8856  cflim2  9029  dfrdg2  31402
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