Theorem ordunisuc 3089

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 2990	. 2 |- (Ord A <-> (A e. On \/ A = On))
2		suceq 3034	. . . . . 6 |- (x = A -> suc x = suc A)
3	2	unieqd 2512	. . . . 5 |- (x = A -> U.suc x = U.suc A)
4		id 59	. . . . 5 |- (x = A -> x = A)
5	3, 4	eqeq12d 1489	. . . 4 |- (x = A -> (U.suc x = x <-> U.suc A = A))
6		eloni 2958	. . . . . 6 |- (x e. On -> Ord x)
7		ordtr 2962	. . . . . 6 |- (Ord x -> Tr x)
8	6, 7	syl 10	. . . . 5 |- (x e. On -> Tr x)
9		visset 1813	. . . . . 6 |- x e. V
10	9	unisuc 3046	. . . . 5 |- (Tr x <-> U.suc x = x)
11	8, 10	sylib 198	. . . 4 |- (x e. On -> U.suc x = x)
12	5, 11	vtoclga 1852	. . 3 |- (A e. On -> U.suc A = A)
13		onprc 2989	. . . . . . 7 |- -. On e. V
14		eleq1 1534	. . . . . . 7 |- (A = On -> (A e. V <-> On e. V))
15	13, 14	mtbiri 717	. . . . . 6 |- (A = On -> -. A e. V)
16		sucprc 3044	. . . . . 6 |- (-. A e. V -> suc A = A)
17	15, 16	syl 10	. . . . 5 |- (A = On -> suc A = A)
18	17	unieqd 2512	. . . 4 |- (A = On -> U.suc A = U.A)
19		unon 3088	. . . . 5 |- U.On = On
20		unieq 2510	. . . . 5 |- (A = On -> U.A = U.On)
21		id 59	. . . . 5 |- (A = On -> A = On)
22	19, 20, 21	3eqtr4a 1532	. . . 4 |- (A = On -> U.A = A)
23	18, 22	eqtrd 1507	. . 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 956   e. wcel 958  Vcvv 1811  U.cuni 2503  Tr wtr 2680  Ord word 2947  Oncon0 2948  suc csuc 2950

This theorem is referenced by:  orduniss2 3090  onsucuni2 3091  nlimsucg 3112

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954

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