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Theorem orduniorsuc 7265
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 6036 . . . . . 6 (Ord 𝐴 𝐴𝐴)
2 orduni 7229 . . . . . . . 8 (Ord 𝐴 → Ord 𝐴)
3 ordelssne 5969 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐴) → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
42, 3mpancom 680 . . . . . . 7 (Ord 𝐴 → ( 𝐴𝐴 ↔ ( 𝐴𝐴 𝐴𝐴)))
54biimprd 240 . . . . . 6 (Ord 𝐴 → (( 𝐴𝐴 𝐴𝐴) → 𝐴𝐴))
61, 5mpand 687 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 𝐴𝐴))
7 ordsucss 7253 . . . . 5 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
86, 7syld 47 . . . 4 (Ord 𝐴 → ( 𝐴𝐴 → suc 𝐴𝐴))
9 ordsucuni 7264 . . . 4 (Ord 𝐴𝐴 ⊆ suc 𝐴)
108, 9jctild 522 . . 3 (Ord 𝐴 → ( 𝐴𝐴 → (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴)))
11 df-ne 2973 . . . 4 (𝐴 𝐴 ↔ ¬ 𝐴 = 𝐴)
12 necom 3025 . . . 4 (𝐴 𝐴 𝐴𝐴)
1311, 12bitr3i 269 . . 3 𝐴 = 𝐴 𝐴𝐴)
14 eqss 3814 . . 3 (𝐴 = suc 𝐴 ↔ (𝐴 ⊆ suc 𝐴 ∧ suc 𝐴𝐴))
1510, 13, 143imtr4g 288 . 2 (Ord 𝐴 → (¬ 𝐴 = 𝐴𝐴 = suc 𝐴))
1615orrd 890 1 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wo 874   = wceq 1653  wcel 2157  wne 2972  wss 3770   cuni 4629  Ord word 5941  suc csuc 5944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-tr 4947  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-ord 5945  df-on 5946  df-suc 5948
This theorem is referenced by:  onuniorsuci  7274  oeeulem  7922  cantnfp1lem2  8827  cantnflem1  8837  cnfcom2lem  8849  dfac12lem1  9254  dfac12lem2  9255  ttukeylem3  9622  ttukeylem5  9624  ttukeylem6  9625  ordtoplem  32941  ordcmp  32953  onsucuni3  33712  aomclem5  38408  onsetreclem3  43247
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