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Theorem ordzsl 7423
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 7421 . . . . . 6 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
21biimprd 249 . . . . 5 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥𝐴 = 𝐴))
3 unizlim 6189 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
42, 3sylibd 240 . . . 4 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴)))
54orrd 858 . . 3 (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
6 3orass 1083 . . . 4 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)))
7 or12 915 . . . 4 ((𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
86, 7bitri 276 . . 3 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
95, 8sylibr 235 . 2 (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
10 ord0 6125 . . . 4 Ord ∅
11 ordeq 6080 . . . 4 (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅))
1210, 11mpbiri 259 . . 3 (𝐴 = ∅ → Ord 𝐴)
13 suceloni 7391 . . . . . 6 (𝑥 ∈ On → suc 𝑥 ∈ On)
14 eleq1 2872 . . . . . 6 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1513, 14syl5ibr 247 . . . . 5 (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On))
16 eloni 6083 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
1715, 16syl6com 37 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴))
1817rexlimiv 3245 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴)
19 limord 6132 . . 3 (Lim 𝐴 → Ord 𝐴)
2012, 18, 193jaoi 1420 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) → Ord 𝐴)
219, 20impbii 210 1 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wo 842  w3o 1079   = wceq 1525  wcel 2083  wrex 3108  c0 4217   cuni 4751  Ord word 6072  Oncon0 6073  Lim wlim 6074  suc csuc 6075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-tr 5071  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079
This theorem is referenced by:  onzsl  7424  tfrlem16  7888  omeulem1  8065  oaabs2  8129  rankxplim3  9163  rankxpsuc  9164  cardlim  9254  cardaleph  9368  cflim2  9538  dfrdg2  32651
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