Theorem omsubsucOLD 15386

Description: The initial ordinal of a successor index is the smallest ordinal strictly larger than the previous initial ordinal. (Moved to omsubsuc 5877 in main set.mm and may be deleted by mathbox owner, JGH. --NM 26-Aug-2011.)

Assertion

Ref	Expression
omsubsucOLD	|- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})

Distinct variable group:   x,A

Proof of Theorem omsubsucOLD

Step	Hyp	Ref	Expression
1		rdgsuc 5153	. . 3 |- (A e. On -> (rec({<.y, z>. | z = |^|{w e. On | y ~< w}}, om)` suc A) = ({<.y, z>. | z = |^|{w e. On | y ~< w}}` (rec({<.y, z>. | z = |^|{w e. On | y ~< w}}, om)` A)))
2		funopabeq 4456	. . . . . 6 |- Fun {<.y, z>. | z = |^|{w e. On | y ~< w}}
3	2	a1i 8	. . . . 5 |- (A e. On -> Fun {<.y, z>. | z = |^|{w e. On | y ~< w}})
4		breq2 3342	. . . . . . . . 9 |- (x = w -> ((aleph` A) ~< x <-> (aleph` A) ~< w))
5	4	cbvrabv 2422	. . . . . . . 8 |- {x e. On | (aleph` A) ~< x} = {w e. On | (aleph` A) ~< w}
6	5	inteqi 3218	. . . . . . 7 |- |^|{x e. On | (aleph` A) ~< x} = |^|{w e. On | (aleph` A) ~< w}
7	6	a1i 8	. . . . . 6 |- (A e. On -> |^|{x e. On | (aleph` A) ~< x} = |^|{w e. On | (aleph` A) ~< w})
8		fvex 4689	. . . . . . 7 |- (aleph` A) e. _V
9		alephon 5876	. . . . . . . . 9 |- (aleph` A) e. On
10		onsdom 5694	. . . . . . . . 9 |- ((aleph` A) e. On -> E.x e. On (aleph` A) ~< x)
11	9, 10	ax-mp 7	. . . . . . . 8 |- E.x e. On (aleph` A) ~< x
12		intexrab 3468	. . . . . . . 8 |- (E.x e. On (aleph` A) ~< x <-> |^|{x e. On | (aleph` A) ~< x} e. _V)
13	11, 12	mpbi 206	. . . . . . 7 |- |^|{x e. On | (aleph` A) ~< x} e. _V
14		breq1 3341	. . . . . . . . . 10 |- (y = (aleph` A) -> (y ~< w <-> (aleph` A) ~< w))
15	14	rabbidv 2287	. . . . . . . . 9 |- (y = (aleph` A) -> {w e. On | y ~< w} = {w e. On | (aleph` A) ~< w})
16	15	inteqd 3219	. . . . . . . 8 |- (y = (aleph` A) -> |^|{w e. On | y ~< w} = |^|{w e. On | (aleph` A) ~< w})
17	16	eqeq2d 1895	. . . . . . 7 |- (y = (aleph` A) -> (z = |^|{w e. On | y ~< w} <-> z = |^|{w e. On | (aleph` A) ~< w}))
18		eqeq1 1890	. . . . . . 7 |- (z = |^|{x e. On | (aleph` A) ~< x} -> (z = |^|{w e. On | (aleph` A) ~< w} <-> |^|{x e. On | (aleph` A) ~< x} = |^|{w e. On | (aleph` A) ~< w}))
19	8, 13, 17, 18	opelopab 3570	. . . . . 6 |- (<.(aleph` A), |^|{x e. On | (aleph` A) ~< x}>. e. {<.y, z>. | z = |^|{w e. On | y ~< w}} <-> |^|{x e. On | (aleph` A) ~< x} = |^|{w e. On | (aleph` A) ~< w})
20	7, 19	sylibr 217	. . . . 5 |- (A e. On -> <.(aleph` A), |^|{x e. On | (aleph` A) ~< x}>. e. {<.y, z>. | z = |^|{w e. On | y ~< w}})
21	13	funopfv 4710	. . . . 5 |- (Fun {<.y, z>. | z = |^|{w e. On | y ~< w}} -> (<.(aleph` A), |^|{x e. On | (aleph` A) ~< x}>. e. {<.y, z>. | z = |^|{w e. On | y ~< w}} -> ({<.y, z>. | z = |^|{w e. On | y ~< w}}` (aleph` A)) = |^|{x e. On | (aleph` A) ~< x}))
22	3, 20, 21	sylc 83	. . . 4 |- (A e. On -> ({<.y, z>. | z = |^|{w e. On | y ~< w}}` (aleph` A)) = |^|{x e. On | (aleph` A) ~< x})
23		df-aleph 5863	. . . . . . 7 |- aleph = rec({<.y, z>. | z = |^|{w e. On | y ~< w}}, om)
24	23	fveq1i 4682	. . . . . 6 |- (aleph` A) = (rec({<.y, z>. | z = |^|{w e. On | y ~< w}}, om)` A)
25	24	fveq2i 4684	. . . . 5 |- ({<.y, z>. | z = |^|{w e. On | y ~< w}}` (aleph` A)) = ({<.y, z>. | z = |^|{w e. On | y ~< w}}` (rec({<.y, z>. | z = |^|{w e. On | y ~< w}}, om)` A))
26	25	eqcomi 1888	. . . 4 |- ({<.y, z>. | z = |^|{w e. On | y ~< w}}` (rec({<.y, z>. | z = |^|{w e. On | y ~< w}}, om)` A)) = ({<.y, z>. | z = |^|{w e. On | y ~< w}}` (aleph` A))
27	22, 26	syl5eq 1940	. . 3 |- (A e. On -> ({<.y, z>. | z = |^|{w e. On | y ~< w}}` (rec({<.y, z>. | z = |^|{w e. On | y ~< w}}, om)` A)) = |^|{x e. On | (aleph` A) ~< x})
28	1, 27	eqtrd 1925	. 2 |- (A e. On -> (rec({<.y, z>. | z = |^|{w e. On | y ~< w}}, om)` suc A) = |^|{x e. On | (aleph` A) ~< x})
29	23	fveq1i 4682	. 2 |- (aleph` suc A) = (rec({<.y, z>. | z = |^|{w e. On | y ~< w}}, om)` suc A)
30	28, 29	syl5eq 1940	1 |- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  E.wrex 2106  {crab 2108  _Vcvv 2292  <.cop 3046  |^|cint 3214   class class class wbr 3338  {copab 3395  Oncon0 3657  suc csuc 3659  omcom 3949  Fun wfun 3992  ` cfv 3998  reccrdg 5139   ~< csdm 5425  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-aleph 5863

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