Theorem omsubinitOLD 15399

Description: An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Moved to omsubinit 5890 in main set.mm and may be deleted by mathbox owner, JGH. --NM 26-Aug-2011.)

Assertion

Ref	Expression
omsubinitOLD	|- ((A e. On /\ om C_ A) -> (A e. ran aleph <-> A.x e. On (A ~<_ x -> A C_ x)))

Distinct variable group:   x,A

Proof of Theorem omsubinitOLD

Step	Hyp	Ref	Expression
1		breq1 3341	. . . . . . . . . . . 12 |- ((aleph` y) = A -> ((aleph` y) ~<_ x <-> A ~<_ x))
2		sseq1 2637	. . . . . . . . . . . 12 |- ((aleph` y) = A -> ((aleph` y) C_ x <-> A C_ x))
3	1, 2	imbi12d 688	. . . . . . . . . . 11 |- ((aleph` y) = A -> (((aleph` y) ~<_ x -> (aleph` y) C_ x) <-> (A ~<_ x -> A C_ x)))
4		omsubdmss 5886	. . . . . . . . . . . 12 |- ((y e. On /\ x e. On) -> ((aleph` y) C_ x <-> (aleph` y) ~<_ x))
5	4	biimprd 171	. . . . . . . . . . 11 |- ((y e. On /\ x e. On) -> ((aleph` y) ~<_ x -> (aleph` y) C_ x))
6	3, 5	syl5cbi 226	. . . . . . . . . 10 |- ((y e. On /\ x e. On) -> ((aleph` y) = A -> (A ~<_ x -> A C_ x)))
7	6	adantl 424	. . . . . . . . 9 |- (((A e. On /\ om C_ A) /\ (y e. On /\ x e. On)) -> ((aleph` y) = A -> (A ~<_ x -> A C_ x)))
8	7	expr 418	. . . . . . . 8 |- (((A e. On /\ om C_ A) /\ y e. On) -> (x e. On -> ((aleph` y) = A -> (A ~<_ x -> A C_ x))))
9	8	com23 36	. . . . . . 7 |- (((A e. On /\ om C_ A) /\ y e. On) -> ((aleph` y) = A -> (x e. On -> (A ~<_ x -> A C_ x))))
10	9	impr 422	. . . . . 6 |- (((A e. On /\ om C_ A) /\ (y e. On /\ (aleph` y) = A)) -> (x e. On -> (A ~<_ x -> A C_ x)))
11	10	r19.21aiv 2175	. . . . 5 |- (((A e. On /\ om C_ A) /\ (y e. On /\ (aleph` y) = A)) -> A.x e. On (A ~<_ x -> A C_ x))
12	11	exp32 408	. . . 4 |- ((A e. On /\ om C_ A) -> (y e. On -> ((aleph` y) = A -> A.x e. On (A ~<_ x -> A C_ x))))
13	12	r19.23adv 2215	. . 3 |- ((A e. On /\ om C_ A) -> (E.y e. On (aleph` y) = A -> A.x e. On (A ~<_ x -> A C_ x)))
14		infenomsub 5889	. . . 4 |- ((A e. On /\ om C_ A) -> E.z e. ran aleph z ~~ A)
15		breq1 3341	. . . . . . . . . 10 |- ((aleph` w) = z -> ((aleph` w) ~~ A <-> z ~~ A))
16	15	imbi1d 675	. . . . . . . . 9 |- ((aleph` w) = z -> (((aleph` w) ~~ A -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A)) <-> (z ~~ A -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A))))
17		simpll 448	. . . . . . . . . . . . . 14 |- (((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) -> A e. On)
18		simprr 451	. . . . . . . . . . . . . 14 |- (((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) -> (aleph` w) ~~ A)
19		ensymg 5470	. . . . . . . . . . . . . 14 |- (A e. On -> ((aleph` w) ~~ A -> A ~~ (aleph` w)))
20	17, 18, 19	sylc 83	. . . . . . . . . . . . 13 |- (((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) -> A ~~ (aleph` w))
21		endom 5444	. . . . . . . . . . . . 13 |- (A ~~ (aleph` w) -> A ~<_ (aleph` w))
22	20, 21	syl 12	. . . . . . . . . . . 12 |- (((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) -> A ~<_ (aleph` w))
23		simplrl 454	. . . . . . . . . . . . . . 15 |- ((((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) /\ A C_ (aleph` w)) -> w e. On)
24		omsubdmss 5886	. . . . . . . . . . . . . . . . . . . . . 22 |- ((w e. On /\ A e. On) -> ((aleph` w) C_ A <-> (aleph` w) ~<_ A))
25	24	ancoms 484	. . . . . . . . . . . . . . . . . . . . 21 |- ((A e. On /\ w e. On) -> ((aleph` w) C_ A <-> (aleph` w) ~<_ A))
26	25	adantlr 429	. . . . . . . . . . . . . . . . . . . 20 |- (((A e. On /\ om C_ A) /\ w e. On) -> ((aleph` w) C_ A <-> (aleph` w) ~<_ A))
27	26	biimprd 171	. . . . . . . . . . . . . . . . . . 19 |- (((A e. On /\ om C_ A) /\ w e. On) -> ((aleph` w) ~<_ A -> (aleph` w) C_ A))
28		endom 5444	. . . . . . . . . . . . . . . . . . 19 |- ((aleph` w) ~~ A -> (aleph` w) ~<_ A)
29	27, 28	syl5 20	. . . . . . . . . . . . . . . . . 18 |- (((A e. On /\ om C_ A) /\ w e. On) -> ((aleph` w) ~~ A -> (aleph` w) C_ A))
30	29	impr 422	. . . . . . . . . . . . . . . . 17 |- (((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) -> (aleph` w) C_ A)
31	30	adantr 425	. . . . . . . . . . . . . . . 16 |- ((((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) /\ A C_ (aleph` w)) -> (aleph` w) C_ A)
32		simpr 350	. . . . . . . . . . . . . . . 16 |- ((((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) /\ A C_ (aleph` w)) -> A C_ (aleph` w))
33	31, 32	eqssd 2633	. . . . . . . . . . . . . . 15 |- ((((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) /\ A C_ (aleph` w)) -> (aleph` w) = A)
34		fveq2 4681	. . . . . . . . . . . . . . . . 17 |- (y = w -> (aleph` y) = (aleph` w))
35	34	eqeq1d 1892	. . . . . . . . . . . . . . . 16 |- (y = w -> ((aleph` y) = A <-> (aleph` w) = A))
36	35	rcla4ev 2381	. . . . . . . . . . . . . . 15 |- ((w e. On /\ (aleph` w) = A) -> E.y e. On (aleph` y) = A)
37	23, 33, 36	syl11anc 524	. . . . . . . . . . . . . 14 |- ((((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) /\ A C_ (aleph` w)) -> E.y e. On (aleph` y) = A)
38	37	ex 402	. . . . . . . . . . . . 13 |- (((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) -> (A C_ (aleph` w) -> E.y e. On (aleph` y) = A))
39	38	imim2d 28	. . . . . . . . . . . 12 |- (((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) -> ((A ~<_ (aleph` w) -> A C_ (aleph` w)) -> (A ~<_ (aleph` w) -> E.y e. On (aleph` y) = A)))
40	22, 39	mpid 58	. . . . . . . . . . 11 |- (((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) -> ((A ~<_ (aleph` w) -> A C_ (aleph` w)) -> E.y e. On (aleph` y) = A))
41		alephon 5876	. . . . . . . . . . . 12 |- (aleph` w) e. On
42		breq2 3342	. . . . . . . . . . . . . 14 |- (x = (aleph` w) -> (A ~<_ x <-> A ~<_ (aleph` w)))
43		sseq2 2639	. . . . . . . . . . . . . 14 |- (x = (aleph` w) -> (A C_ x <-> A C_ (aleph` w)))
44	42, 43	imbi12d 688	. . . . . . . . . . . . 13 |- (x = (aleph` w) -> ((A ~<_ x -> A C_ x) <-> (A ~<_ (aleph` w) -> A C_ (aleph` w))))
45	44	rcla4v 2376	. . . . . . . . . . . 12 |- ((aleph` w) e. On -> (A.x e. On (A ~<_ x -> A C_ x) -> (A ~<_ (aleph` w) -> A C_ (aleph` w))))
46	41, 45	ax-mp 7	. . . . . . . . . . 11 |- (A.x e. On (A ~<_ x -> A C_ x) -> (A ~<_ (aleph` w) -> A C_ (aleph` w)))
47	40, 46	syl5 20	. . . . . . . . . 10 |- (((A e. On /\ om C_ A) /\ (w e. On /\ (aleph` w) ~~ A)) -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A))
48	47	expr 418	. . . . . . . . 9 |- (((A e. On /\ om C_ A) /\ w e. On) -> ((aleph` w) ~~ A -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A)))
49	16, 48	syl5cbi 226	. . . . . . . 8 |- (((A e. On /\ om C_ A) /\ w e. On) -> ((aleph` w) = z -> (z ~~ A -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A))))
50	49	ex 402	. . . . . . 7 |- ((A e. On /\ om C_ A) -> (w e. On -> ((aleph` w) = z -> (z ~~ A -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A)))))
51	50	r19.23adv 2215	. . . . . 6 |- ((A e. On /\ om C_ A) -> (E.w e. On (aleph` w) = z -> (z ~~ A -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A))))
52		alephfnon 5873	. . . . . . 7 |- aleph Fn On
53		fvelrnb 4719	. . . . . . 7 |- (aleph Fn On -> (z e. ran aleph <-> E.w e. On (aleph` w) = z))
54	52, 53	ax-mp 7	. . . . . 6 |- (z e. ran aleph <-> E.w e. On (aleph` w) = z)
55	51, 54	syl5ib 223	. . . . 5 |- ((A e. On /\ om C_ A) -> (z e. ran aleph -> (z ~~ A -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A))))
56	55	r19.23adv 2215	. . . 4 |- ((A e. On /\ om C_ A) -> (E.z e. ran aleph z ~~ A -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A)))
57	14, 56	mpd 29	. . 3 |- ((A e. On /\ om C_ A) -> (A.x e. On (A ~<_ x -> A C_ x) -> E.y e. On (aleph` y) = A))
58	13, 57	impbid 574	. 2 |- ((A e. On /\ om C_ A) -> (E.y e. On (aleph` y) = A <-> A.x e. On (A ~<_ x -> A C_ x)))
59		fvelrnb 4719	. . 3 |- (aleph Fn On -> (A e. ran aleph <-> E.y e. On (aleph` y) = A))
60	52, 59	ax-mp 7	. 2 |- (A e. ran aleph <-> E.y e. On (aleph` y) = A)
61	58, 60	syl5bb 591	1 |- ((A e. On /\ om C_ A) -> (A e. ran aleph <-> A.x e. On (A ~<_ x -> A C_ x)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593   class class class wbr 3338  Oncon0 3657  omcom 3949  ran crn 3987   Fn wfn 3993  ` cfv 3998   ~~ cen 5423   ~<_ cdom 5424  alephcale 5860

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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-iso 4015  df-oprab 4887  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-aleph 5863

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