Theorem orduniorsuc 2337

Description: An ordinal class is either its union or the successor of its union.

Assertion

Ref	Expression
orduniorsuc	⊢ (Ord A → (A = ∪A ∨ A = suc ∪A))

Proof of Theorem orduniorsuc

Step	Hyp	Ref	Expression
1		ordsucuni 2336	. . . . 5 ⊢ (Ord A → A ⊆ suc ∪A)
2	1	a1d 14	. . . 4 ⊢ (Ord A → (¬ ∪A = A → A ⊆ suc ∪A))
3		orduniss 2327	. . . . . 6 ⊢ (Ord A → ∪A ⊆ A)
4		uniord 2252	. . . . . . . 8 ⊢ (Ord A → Ord ∪A)
5		ordelssne 2225	. . . . . . . 8 ⊢ ((Ord ∪A ∧ Ord A) → (∪A ∈ A ↔ (∪A ⊆ A ∧ ¬ ∪A = A)))
6	4, 5	mpancom 528	. . . . . . 7 ⊢ (Ord A → (∪A ∈ A ↔ (∪A ⊆ A ∧ ¬ ∪A = A)))
7	6	biimprd 136	. . . . . 6 ⊢ (Ord A → ((∪A ⊆ A ∧ ¬ ∪A = A) → ∪A ∈ A))
8	3, 7	mpand 524	. . . . 5 ⊢ (Ord A → (¬ ∪A = A → ∪A ∈ A))
9		ordsucss 2320	. . . . 5 ⊢ (Ord A → (∪A ∈ A → suc ∪A ⊆ A))
10	8, 9	syld 27	. . . 4 ⊢ (Ord A → (¬ ∪A = A → suc ∪A ⊆ A))
11	2, 10	jcad 455	. . 3 ⊢ (Ord A → (¬ ∪A = A → (A ⊆ suc ∪A ∧ suc ∪A ⊆ A)))
12		cleqcom 1103	. . . 4 ⊢ (A = ∪A ↔ ∪A = A)
13	12	negbii 162	. . 3 ⊢ (¬ A = ∪A ↔ ¬ ∪A = A)
14		eqss 1516	. . 3 ⊢ (A = suc ∪A ↔ (A ⊆ suc ∪A ∧ suc ∪A ⊆ A))
15	11, 13, 14	3imtr4g 426	. 2 ⊢ (Ord A → (¬ A = ∪A → A = suc ∪A))
16	15	orrd 203	1 ⊢ (Ord A → (A = ∪A ∨ A = suc ∪A))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ∪cuni 1919  Ord word 2198  suc csuc 2201

This theorem is referenced by:  onuniorsuc 2355

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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