Theorem ordunisuc 3911

Description: An ordinal class is equal to the union of its successor. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ordunisuc	|- (Ord A -> U.suc A = A)

Proof of Theorem ordunisuc

Step	Hyp	Ref	Expression
1		ordeleqon 3866	. 2 |- (Ord A <-> (A e. On \/ A = On))
2		suceq 3729	. . . . . 6 |- (x = A -> suc x = suc A)
3	2	unieqd 3188	. . . . 5 |- (x = A -> U.suc x = U.suc A)
4		id 73	. . . . 5 |- (x = A -> x = A)
5	3, 4	eqeq12d 1899	. . . 4 |- (x = A -> (U.suc x = x <-> U.suc A = A))
6		eloni 3667	. . . . . 6 |- (x e. On -> Ord x)
7		ordtr 3672	. . . . . 6 |- (Ord x -> Tr x)
8	6, 7	syl 12	. . . . 5 |- (x e. On -> Tr x)
9		visset 2295	. . . . . 6 |- x e. _V
10	9	unisuc 3741	. . . . 5 |- (Tr x <-> U.suc x = x)
11	8, 10	sylib 215	. . . 4 |- (x e. On -> U.suc x = x)
12	5, 11	vtoclga 2352	. . 3 |- (A e. On -> U.suc A = A)
13		sucon 3889	. . . . . 6 |- suc On = On
14	13	unieqi 3187	. . . . 5 |- U.suc On = U.On
15		unon 3910	. . . . 5 |- U.On = On
16	14, 15	eqtri 1908	. . . 4 |- U.suc On = On
17		suceq 3729	. . . . 5 |- (A = On -> suc A = suc On)
18	17	unieqd 3188	. . . 4 |- (A = On -> U.suc A = U.suc On)
19		id 73	. . . 4 |- (A = On -> A = On)
20	16, 18, 19	3eqtr4a 1954	. . 3 |- (A = On -> U.suc A = A)
21	12, 20	jaoi 368	. 2 |- ((A e. On \/ A = On) -> U.suc A = A)
22	1, 21	sylbi 216	1 |- (Ord A -> U.suc A = A)

Syntax hints:   -> wi 3   \/ wo 239   = wceq 1298   e. wcel 1300  U.cuni 3177  Tr wtr 3411  Ord word 3656  Oncon0 3657  suc csuc 3659

This theorem is referenced by:  orduniss2 3913  onsucuni2 3914  onsucuni2OLD 3915  nlimsucg 3923  nlimsucgOLD 3924  ordsucuniel 13863

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663

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