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Theorem orduniorsuc 7820
Description: An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.)
Assertion
Ref Expression
orduniorsuc (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))

Proof of Theorem orduniorsuc
StepHypRef Expression
1 orduniss 6460 . . . . . 6 (Ord 𝐴 𝐴𝐴)
2 orduni 7779 . . . . . . . 8 (Ord 𝐴 → Ord 𝐴)
3 ordelssne 6390 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐴) → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
42, 3mpancom 684 . . . . . . 7 (Ord 𝐴 → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
54biimprd 247 . . . . . 6 (Ord 𝐴 → (( 𝐴𝐴 𝐴𝐴) → 𝐴𝐴))
61, 5mpand 691 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 𝐴𝐴))
7 ordsucss 7808 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
86, 7syld 47 . . . 4 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
9 ordsucuni 7819 . . . 4 (Ord 𝐴𝐴 ⊆ suc 𝐴)
108, 9jctild 524 . . 3 (Ord 𝐴 → ( 𝐴𝐴 → (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴)))
11 df-ne 2939 . . . 4 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
12 necom 2992 . . . 4 (𝐴 𝐴 𝐴𝐴)
1311, 12bitr3i 276 . . 3 𝐴 = 𝐴 𝐴𝐴)
14 eqss 3996 . . 3 (𝐴 = suc 𝐴 ↔ (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴))
1510, 13, 143imtr4g 295 . 2 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
1615orrd 859 1 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 843   = wceq 1539  wcel 2104  wne 2938  wss 3947   cuni 4907  Ord word 6362  suc csuc 6365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366  df-on 6367  df-suc 6369
This theorem is referenced by:  onuniorsuc  7827  oeeulem  8603  cantnfp1lem2  9676  cantnflem1  9686  cnfcom2lem  9698  dfac12lem1  10140  dfac12lem2  10141  ttukeylem3  10508  ttukeylem5  10510  ttukeylem6  10511  ordtoplem  35623  ordcmp  35635  onsucuni3  36551  aomclem5  42102  omlimcl2  42293  onov0suclim  42326  dflim5  42381  onsetreclem3  47839
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