Theorem ordunisuc 2339

Description: An ordinal class is equal to the union of its successor.

Assertion

Ref	Expression
ordunisuc	⊢ (Ord A → ∪suc A = A)

Proof of Theorem ordunisuc

Step	Hyp	Ref	Expression
1		ordeleqon 2241	. 2 ⊢ (Ord A ↔ (A ∈ On ∨ A = On))
2		suceq 2288	. . . . . 6 ⊢ (x = A → suc x = suc A)
3	2	unieqd 1929	. . . . 5 ⊢ (x = A → ∪suc x = ∪suc A)
4		id 9	. . . . 5 ⊢ (x = A → x = A)
5	3, 4	cleq12d 1115	. . . 4 ⊢ (x = A → (∪suc x = x ↔ ∪suc A = A))
6		eloni 2209	. . . . . 6 ⊢ (x ∈ On → Ord x)
7		ordtr 2213	. . . . . 6 ⊢ (Ord x → Tr x)
8	6, 7	syl 12	. . . . 5 ⊢ (x ∈ On → Tr x)
9		visset 1350	. . . . . 6 ⊢ x ∈ V
10	9	unisuc 2299	. . . . 5 ⊢ (Tr x ↔ ∪suc x = x)
11	8, 10	sylib 173	. . . 4 ⊢ (x ∈ On → ∪suc x = x)
12	5, 11	vtoclga 1387	. . 3 ⊢ (A ∈ On → ∪suc A = A)
13		onprc 2240	. . . . . . 7 ⊢ ¬ On ∈ V
14		eleq1 1149	. . . . . . 7 ⊢ (A = On → (A ∈ V ↔ On ∈ V))
15	13, 14	mtbiri 539	. . . . . 6 ⊢ (A = On → ¬ A ∈ V)
16		sucprc 2297	. . . . . 6 ⊢ (¬ A ∈ V → suc A = A)
17	15, 16	syl 12	. . . . 5 ⊢ (A = On → suc A = A)
18	17	unieqd 1929	. . . 4 ⊢ (A = On → ∪suc A = ∪A)
19		unon 2338	. . . . 5 ⊢ ∪On = On
20		unieq 1927	. . . . . 6 ⊢ (A = On → ∪A = ∪On)
21		id 9	. . . . . 6 ⊢ (A = On → A = On)
22	20, 21	cleq12d 1115	. . . . 5 ⊢ (A = On → (∪A = A ↔ ∪On = On))
23	19, 22	mpbiri 169	. . . 4 ⊢ (A = On → ∪A = A)
24	18, 23	eqtrd 1128	. . 3 ⊢ (A = On → ∪suc A = A)
25	12, 24	jaoi 275	. 2 ⊢ ((A ∈ On ∨ A = On) → ∪suc A = A)
26	1, 25	sylbi 174	1 ⊢ (Ord A → ∪suc A = A)

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∪cuni 1919  Tr wtr 2041  Ord word 2198  Oncon0 2199  suc csuc 2201

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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