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Theorem orduniorsuc 7849
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 6482 . . . . . 6 (Ord 𝐴 𝐴𝐴)
2 orduni 7808 . . . . . . . 8 (Ord 𝐴 → Ord 𝐴)
3 ordelssne 6412 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐴) → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
42, 3mpancom 688 . . . . . . 7 (Ord 𝐴 → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
54biimprd 248 . . . . . 6 (Ord 𝐴 → (( 𝐴𝐴 𝐴𝐴) → 𝐴𝐴))
61, 5mpand 695 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 𝐴𝐴))
7 ordsucss 7837 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
86, 7syld 47 . . . 4 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
9 ordsucuni 7848 . . . 4 (Ord 𝐴𝐴 ⊆ suc 𝐴)
108, 9jctild 525 . . 3 (Ord 𝐴 → ( 𝐴𝐴 → (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴)))
11 df-ne 2938 . . . 4 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
12 necom 2991 . . . 4 (𝐴 𝐴 𝐴𝐴)
1311, 12bitr3i 277 . . 3 𝐴 = 𝐴 𝐴𝐴)
14 eqss 4010 . . 3 (𝐴 = suc 𝐴 ↔ (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴))
1510, 13, 143imtr4g 296 . 2 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
1615orrd 863 1 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937  wss 3962   cuni 4911  Ord word 6384  suc csuc 6387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-suc 6391
This theorem is referenced by:  onuniorsuc  7856  oeeulem  8637  cantnfp1lem2  9716  cantnflem1  9726  cnfcom2lem  9738  dfac12lem1  10181  dfac12lem2  10182  ttukeylem3  10548  ttukeylem5  10550  ttukeylem6  10551  ordtoplem  36417  ordcmp  36429  onsucuni3  37349  aomclem5  43046  omlimcl2  43230  onov0suclim  43263  dflim5  43318  onsetreclem3  48937
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