Theorem ordunisuc2 3926

Description: An ordinal equal to its union contains the successor of each of its members.

Assertion

Ref	Expression
ordunisuc2	|- (Ord A -> (A = U.A <-> A.x e. A suc x e. A))

Distinct variable group:   x,A

Proof of Theorem ordunisuc2

Step	Hyp	Ref	Expression
1		orduninsuc 3925	. 2 |- (Ord A -> (A = U.A <-> -. E.x e. On A = suc x))
2		ordtri3 3697	. . . . . . . . 9 |- ((Ord A /\ Ord suc x) -> (A = suc x <-> -. (A e. suc x \/ suc x e. A)))
3		suceloni 3894	. . . . . . . . . 10 |- (x e. On -> suc x e. On)
4		eloni 3667	. . . . . . . . . 10 |- (suc x e. On -> Ord suc x)
5	3, 4	syl 12	. . . . . . . . 9 |- (x e. On -> Ord suc x)
6	2, 5	sylan2 500	. . . . . . . 8 |- ((Ord A /\ x e. On) -> (A = suc x <-> -. (A e. suc x \/ suc x e. A)))
7	6	con2bid 585	. . . . . . 7 |- ((Ord A /\ x e. On) -> ((A e. suc x \/ suc x e. A) <-> -. A = suc x))
8		onnbtwn 3762	. . . . . . . . . . . . 13 |- (x e. On -> -. (x e. A /\ A e. suc x))
9		imnan 261	. . . . . . . . . . . . 13 |- ((x e. A -> -. A e. suc x) <-> -. (x e. A /\ A e. suc x))
10	8, 9	sylibr 217	. . . . . . . . . . . 12 |- (x e. On -> (x e. A -> -. A e. suc x))
11	10	con2d 107	. . . . . . . . . . 11 |- (x e. On -> (A e. suc x -> -. x e. A))
12		pm2.21 92	. . . . . . . . . . 11 |- (-. x e. A -> (x e. A -> suc x e. A))
13	11, 12	syl6 25	. . . . . . . . . 10 |- (x e. On -> (A e. suc x -> (x e. A -> suc x e. A)))
14	13	adantl 424	. . . . . . . . 9 |- ((Ord A /\ x e. On) -> (A e. suc x -> (x e. A -> suc x e. A)))
15		ax-1 4	. . . . . . . . . 10 |- (suc x e. A -> (x e. A -> suc x e. A))
16	15	a1i 8	. . . . . . . . 9 |- ((Ord A /\ x e. On) -> (suc x e. A -> (x e. A -> suc x e. A)))
17	14, 16	jaod 469	. . . . . . . 8 |- ((Ord A /\ x e. On) -> ((A e. suc x \/ suc x e. A) -> (x e. A -> suc x e. A)))
18		ordtri2or 3766	. . . . . . . . . . . . . 14 |- ((Ord x /\ Ord A) -> (x e. A \/ A C_ x))
19		eloni 3667	. . . . . . . . . . . . . 14 |- (x e. On -> Ord x)
20	18, 19	sylan 497	. . . . . . . . . . . . 13 |- ((x e. On /\ Ord A) -> (x e. A \/ A C_ x))
21	20	ancoms 484	. . . . . . . . . . . 12 |- ((Ord A /\ x e. On) -> (x e. A \/ A C_ x))
22		orcom 266	. . . . . . . . . . . 12 |- ((x e. A \/ A C_ x) <-> (A C_ x \/ x e. A))
23	21, 22	sylib 215	. . . . . . . . . . 11 |- ((Ord A /\ x e. On) -> (A C_ x \/ x e. A))
24	23	adantr 425	. . . . . . . . . 10 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (A C_ x \/ x e. A))
25		ordsssuc2 3758	. . . . . . . . . . . . 13 |- ((Ord A /\ x e. On) -> (A C_ x <-> A e. suc x))
26	25	biimpd 170	. . . . . . . . . . . 12 |- ((Ord A /\ x e. On) -> (A C_ x -> A e. suc x))
27	26	adantr 425	. . . . . . . . . . 11 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (A C_ x -> A e. suc x))
28		simpr 350	. . . . . . . . . . 11 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (x e. A -> suc x e. A))
29	27, 28	orim12d 624	. . . . . . . . . 10 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> ((A C_ x \/ x e. A) -> (A e. suc x \/ suc x e. A)))
30	24, 29	mpd 29	. . . . . . . . 9 |- (((Ord A /\ x e. On) /\ (x e. A -> suc x e. A)) -> (A e. suc x \/ suc x e. A))
31	30	ex 402	. . . . . . . 8 |- ((Ord A /\ x e. On) -> ((x e. A -> suc x e. A) -> (A e. suc x \/ suc x e. A)))
32	17, 31	impbid 574	. . . . . . 7 |- ((Ord A /\ x e. On) -> ((A e. suc x \/ suc x e. A) <-> (x e. A -> suc x e. A)))
33	7, 32	bitr3d 589	. . . . . 6 |- ((Ord A /\ x e. On) -> (-. A = suc x <-> (x e. A -> suc x e. A)))
34	33	pm5.74da 646	. . . . 5 |- (Ord A -> ((x e. On -> -. A = suc x) <-> (x e. On -> (x e. A -> suc x e. A))))
35		simpr 350	. . . . . . . 8 |- ((x e. On /\ x e. A) -> x e. A)
36		ordelon 3682	. . . . . . . . . 10 |- ((Ord A /\ x e. A) -> x e. On)
37	36	ex 402	. . . . . . . . 9 |- (Ord A -> (x e. A -> x e. On))
38	37	ancrd 323	. . . . . . . 8 |- (Ord A -> (x e. A -> (x e. On /\ x e. A)))
39	35, 38	impbid2 576	. . . . . . 7 |- (Ord A -> ((x e. On /\ x e. A) <-> x e. A))
40	39	imbi1d 675	. . . . . 6 |- (Ord A -> (((x e. On /\ x e. A) -> suc x e. A) <-> (x e. A -> suc x e. A)))
41		impexp 374	. . . . . 6 |- (((x e. On /\ x e. A) -> suc x e. A) <-> (x e. On -> (x e. A -> suc x e. A)))
42	40, 41	syl5bbr 593	. . . . 5 |- (Ord A -> ((x e. On -> (x e. A -> suc x e. A)) <-> (x e. A -> suc x e. A)))
43	34, 42	bitrd 587	. . . 4 |- (Ord A -> ((x e. On -> -. A = suc x) <-> (x e. A -> suc x e. A)))
44	43	ralbidv2 2125	. . 3 |- (Ord A -> (A.x e. On -. A = suc x <-> A.x e. A suc x e. A))
45		ralnex 2113	. . 3 |- (A.x e. On -. A = suc x <-> -. E.x e. On A = suc x)
46	44, 45	syl5bbr 593	. 2 |- (Ord A -> (-. E.x e. On A = suc x <-> A.x e. A suc x e. A))
47	1, 46	bitrd 587	1 |- (Ord A -> (A = U.A <-> A.x e. A suc x e. A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  U.cuni 3177  Ord word 3656  Oncon0 3657  suc csuc 3659

This theorem is referenced by:  dflim4 3932

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-if 2983  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

Copyright terms: Public domain