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Theorem orduniorsuc 6922
 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 5738 . . . . . 6 (Ord 𝐴 𝐴𝐴)
2 orduni 6886 . . . . . . . 8 (Ord 𝐴 → Ord 𝐴)
3 ordelssne 5667 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐴) → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
42, 3mpancom 700 . . . . . . 7 (Ord 𝐴 → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
54biimprd 237 . . . . . 6 (Ord 𝐴 → (( 𝐴𝐴 𝐴𝐴) → 𝐴𝐴))
61, 5mpand 707 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 𝐴𝐴))
7 ordsucss 6910 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
86, 7syld 46 . . . 4 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
9 ordsucuni 6921 . . . 4 (Ord 𝐴𝐴 ⊆ suc 𝐴)
108, 9jctild 564 . . 3 (Ord 𝐴 → ( 𝐴𝐴 → (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴)))
11 df-ne 2782 . . . 4 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
12 necom 2835 . . . 4 (𝐴 𝐴 𝐴𝐴)
1311, 12bitr3i 265 . . 3 𝐴 = 𝐴 𝐴𝐴)
14 eqss 3583 . . 3 (𝐴 = suc 𝐴 ↔ (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴))
1510, 13, 143imtr4g 284 . 2 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
1615orrd 392 1 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∪ cuni 4372  Ord word 5639  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646 This theorem is referenced by:  onuniorsuci  6931  oeeulem  7568  cantnfp1lem2  8459  cantnflem1  8469  cnfcom2lem  8481  dfac12lem1  8848  dfac12lem2  8849  ttukeylem3  9216  ttukeylem5  9218  ttukeylem6  9219  ordtoplem  31604  ordcmp  31616  onsucuni3  32391  aomclem5  36646  onsetreclem3  42249
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