Theorem ordunisuc 3085

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

Assertion

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

Proof of Theorem ordunisuc

Step	Hyp	Ref	Expression
1		ordeleqon 2986	. 2 |- (Ord A <-> (A e. On \/ A = On))
2		suceq 3030	. . . . . 6 |- (x = A -> suc x = suc A)
3	2	unieqd 2508	. . . . 5 |- (x = A -> U.suc x = U.suc A)
4		id 59	. . . . 5 |- (x = A -> x = A)
5	3, 4	eqeq12d 1487	. . . 4 |- (x = A -> (U.suc x = x <-> U.suc A = A))
6		eloni 2954	. . . . . 6 |- (x e. On -> Ord x)
7		ordtr 2958	. . . . . 6 |- (Ord x -> Tr x)
8	6, 7	syl 10	. . . . 5 |- (x e. On -> Tr x)
9		visset 1810	. . . . . 6 |- x e. V
10	9	unisuc 3042	. . . . 5 |- (Tr x <-> U.suc x = x)
11	8, 10	sylib 198	. . . 4 |- (x e. On -> U.suc x = x)
12	5, 11	vtoclga 1849	. . 3 |- (A e. On -> U.suc A = A)
13		onprc 2985	. . . . . . 7 |- -. On e. V
14		eleq1 1532	. . . . . . 7 |- (A = On -> (A e. V <-> On e. V))
15	13, 14	mtbiri 716	. . . . . 6 |- (A = On -> -. A e. V)
16		sucprc 3040	. . . . . 6 |- (-. A e. V -> suc A = A)
17	15, 16	syl 10	. . . . 5 |- (A = On -> suc A = A)
18	17	unieqd 2508	. . . 4 |- (A = On -> U.suc A = U.A)
19		unon 3084	. . . . 5 |- U.On = On
20		unieq 2506	. . . . 5 |- (A = On -> U.A = U.On)
21		id 59	. . . . 5 |- (A = On -> A = On)
22	19, 20, 21	3eqtr4a 1530	. . . 4 |- (A = On -> U.A = A)
23	18, 22	eqtrd 1505	. . 3 |- (A = On -> U.suc A = A)
24	12, 23	jaoi 341	. 2 |- ((A e. On \/ A = On) -> U.suc A = A)
25	1, 24	sylbi 199	1 |- (Ord A -> U.suc A = A)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   = wceq 955   e. wcel 957  Vcvv 1808  U.cuni 2499  Tr wtr 2676  Ord word 2943  Oncon0 2944  suc csuc 2946

This theorem is referenced by:  orduniss2 3086  onsucuni2 3087  nlimsucg 3108

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950

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