Theorem orduniorsuc 7539

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 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))

Proof of Theorem orduniorsuc

Step	Hyp	Ref	Expression
1		orduniss 6280	. . . . . 6 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴)
2		orduni 7503	. . . . . . . 8 ⊢ (Ord 𝐴 → Ord ∪ 𝐴)
3		ordelssne 6213	. . . . . . . 8 ⊢ ((Ord ∪ 𝐴 ∧ Ord 𝐴) → (∪ 𝐴 ∈ 𝐴 ↔ (∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴)))
4	2, 3	mpancom 686	. . . . . . 7 ⊢ (Ord 𝐴 → (∪ 𝐴 ∈ 𝐴 ↔ (∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴)))
5	4	biimprd 250	. . . . . 6 ⊢ (Ord 𝐴 → ((∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴) → ∪ 𝐴 ∈ 𝐴))
6	1, 5	mpand 693	. . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → ∪ 𝐴 ∈ 𝐴))
7		ordsucss 7527	. . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 ∈ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴))
8	6, 7	syld 47	. . . 4 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴))
9		ordsucuni 7538	. . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ suc ∪ 𝐴)
10	8, 9	jctild 528	. . 3 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → (𝐴 ⊆ suc ∪ 𝐴 ∧ suc ∪ 𝐴 ⊆ 𝐴)))
11		df-ne 3017	. . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴)
12		necom 3069	. . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴)
13	11, 12	bitr3i 279	. . 3 ⊢ (¬ 𝐴 = ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴)
14		eqss 3982	. . 3 ⊢ (𝐴 = suc ∪ 𝐴 ↔ (𝐴 ⊆ suc ∪ 𝐴 ∧ suc ∪ 𝐴 ⊆ 𝐴))
15	10, 13, 14	3imtr4g 298	. 2 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
16	15	orrd 859	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   ⊆ wss 3936  ∪ cuni 4832  Ord word 6185  suc csuc 6188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-tr 5166  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-ord 6189  df-on 6190  df-suc 6192

This theorem is referenced by:  onuniorsuci  7548  oeeulem  8221  cantnfp1lem2  9136  cantnflem1  9146  cnfcom2lem  9158  dfac12lem1  9563  dfac12lem2  9564  ttukeylem3  9927  ttukeylem5  9929  ttukeylem6  9930  ordtoplem  33778  ordcmp  33790  onsucuni3  34642  aomclem5  39651  onsetreclem3  44802