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Theorem orduniorsuc 7770
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 6419 . . . . . 6 (Ord 𝐴 𝐴𝐴)
2 orduni 7729 . . . . . . . 8 (Ord 𝐴 → Ord 𝐴)
3 ordelssne 6349 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐴) → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
42, 3mpancom 687 . . . . . . 7 (Ord 𝐴 → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
54biimprd 248 . . . . . 6 (Ord 𝐴 → (( 𝐴𝐴 𝐴𝐴) → 𝐴𝐴))
61, 5mpand 694 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 𝐴𝐴))
7 ordsucss 7758 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
86, 7syld 47 . . . 4 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
9 ordsucuni 7769 . . . 4 (Ord 𝐴𝐴 ⊆ suc 𝐴)
108, 9jctild 527 . . 3 (Ord 𝐴 → ( 𝐴𝐴 → (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴)))
11 df-ne 2945 . . . 4 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
12 necom 2998 . . . 4 (𝐴 𝐴 𝐴𝐴)
1311, 12bitr3i 277 . . 3 𝐴 = 𝐴 𝐴𝐴)
14 eqss 3964 . . 3 (𝐴 = suc 𝐴 ↔ (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴))
1510, 13, 143imtr4g 296 . 2 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
1615orrd 862 1 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2944  wss 3915   cuni 4870  Ord word 6321  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  onuniorsuc  7777  oeeulem  8553  cantnfp1lem2  9622  cantnflem1  9632  cnfcom2lem  9644  dfac12lem1  10086  dfac12lem2  10087  ttukeylem3  10454  ttukeylem5  10456  ttukeylem6  10457  ordtoplem  34936  ordcmp  34948  onsucuni3  35867  aomclem5  41414  omlimcl2  41605  onov0suclim  41638  dflim5  41693  onsetreclem3  47226
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