Theorem omsubsuc2 5878

Description: An initial ordinal with successor index. (Contributed by Jeff Hankins, 26-Oct-2009.)

Assertion

Ref	Expression
omsubsuc2	|- (A e. On -> (aleph` suc A) = {x e. On | x ~<_ (aleph` A)})

Distinct variable group:   x,A

Proof of Theorem omsubsuc2

Step	Hyp	Ref	Expression
1		omsubsuc 5877	. 2 |- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})
2		alephon 5876	. . . . . . . . . . 11 |- (aleph` A) e. On
3		onsdom 5694	. . . . . . . . . . 11 |- ((aleph` A) e. On -> E.x e. On (aleph` A) ~< x)
4	2, 3	ax-mp 7	. . . . . . . . . 10 |- E.x e. On (aleph` A) ~< x
5		onintrab2 3883	. . . . . . . . . 10 |- (E.x e. On (aleph` A) ~< x <-> |^|{x e. On | (aleph` A) ~< x} e. On)
6	4, 5	mpbi 206	. . . . . . . . 9 |- |^|{x e. On | (aleph` A) ~< x} e. On
7	6	a1i 8	. . . . . . . 8 |- ((A e. On /\ y e. |^|{x e. On | (aleph` A) ~< x}) -> |^|{x e. On | (aleph` A) ~< x} e. On)
8		onelon 3683	. . . . . . . 8 |- ((|^|{x e. On | (aleph` A) ~< x} e. On /\ y e. |^|{x e. On | (aleph` A) ~< x}) -> y e. On)
9	7, 8	sylancom 531	. . . . . . 7 |- ((A e. On /\ y e. |^|{x e. On | (aleph` A) ~< x}) -> y e. On)
10		breq2 3342	. . . . . . . . . . . . 13 |- (x = y -> ((aleph` A) ~< x <-> (aleph` A) ~< y))
11	10	onnminsb 3885	. . . . . . . . . . . 12 |- (y e. On -> (y e. |^|{x e. On | (aleph` A) ~< x} -> -. (aleph` A) ~< y))
12	11	adantl 424	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (y e. |^|{x e. On | (aleph` A) ~< x} -> -. (aleph` A) ~< y))
13		domtriord 5546	. . . . . . . . . . . . 13 |- ((y e. On /\ (aleph` A) e. On) -> (y ~<_ (aleph` A) <-> -. (aleph` A) ~< y))
14	2, 13	mpan2 760	. . . . . . . . . . . 12 |- (y e. On -> (y ~<_ (aleph` A) <-> -. (aleph` A) ~< y))
15	14	adantl 424	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (y ~<_ (aleph` A) <-> -. (aleph` A) ~< y))
16	12, 15	sylibrd 221	. . . . . . . . . 10 |- ((A e. On /\ y e. On) -> (y e. |^|{x e. On | (aleph` A) ~< x} -> y ~<_ (aleph` A)))
17	16	ex 402	. . . . . . . . 9 |- (A e. On -> (y e. On -> (y e. |^|{x e. On | (aleph` A) ~< x} -> y ~<_ (aleph` A))))
18	17	com23 36	. . . . . . . 8 |- (A e. On -> (y e. |^|{x e. On | (aleph` A) ~< x} -> (y e. On -> y ~<_ (aleph` A))))
19	18	imp 377	. . . . . . 7 |- ((A e. On /\ y e. |^|{x e. On | (aleph` A) ~< x}) -> (y e. On -> y ~<_ (aleph` A)))
20	9, 19	jcai 313	. . . . . 6 |- ((A e. On /\ y e. |^|{x e. On | (aleph` A) ~< x}) -> (y e. On /\ y ~<_ (aleph` A)))
21	20	ex 402	. . . . 5 |- (A e. On -> (y e. |^|{x e. On | (aleph` A) ~< x} -> (y e. On /\ y ~<_ (aleph` A))))
22		sdomirr 5535	. . . . . . . . . . 11 |- -. (aleph` A) ~< (aleph` A)
23		simprlr 457	. . . . . . . . . . . . 13 |- (((A e. On /\ (y e. On /\ y ~<_ (aleph` A))) /\ ((z e. On /\ (aleph` A) ~< z) /\ z C_ y)) -> (aleph` A) ~< z)
24		visset 2295	. . . . . . . . . . . . . . . 16 |- z e. _V
25		ssdomg 5467	. . . . . . . . . . . . . . . 16 |- (z e. _V -> (z C_ y -> z ~<_ y))
26	24, 25	ax-mp 7	. . . . . . . . . . . . . . 15 |- (z C_ y -> z ~<_ y)
27	26	ad2antll 443	. . . . . . . . . . . . . 14 |- (((A e. On /\ (y e. On /\ y ~<_ (aleph` A))) /\ ((z e. On /\ (aleph` A) ~< z) /\ z C_ y)) -> z ~<_ y)
28		simplrr 455	. . . . . . . . . . . . . 14 |- (((A e. On /\ (y e. On /\ y ~<_ (aleph` A))) /\ ((z e. On /\ (aleph` A) ~< z) /\ z C_ y)) -> y ~<_ (aleph` A))
29		domtr 5474	. . . . . . . . . . . . . 14 |- ((z ~<_ y /\ y ~<_ (aleph` A)) -> z ~<_ (aleph` A))
30	27, 28, 29	syl11anc 524	. . . . . . . . . . . . 13 |- (((A e. On /\ (y e. On /\ y ~<_ (aleph` A))) /\ ((z e. On /\ (aleph` A) ~< z) /\ z C_ y)) -> z ~<_ (aleph` A))
31		fvex 4689	. . . . . . . . . . . . . 14 |- (aleph` A) e. _V
32		sdomdomtr 5532	. . . . . . . . . . . . . 14 |- ((aleph` A) e. _V -> (((aleph` A) ~< z /\ z ~<_ (aleph` A)) -> (aleph` A) ~< (aleph` A)))
33	31, 32	ax-mp 7	. . . . . . . . . . . . 13 |- (((aleph` A) ~< z /\ z ~<_ (aleph` A)) -> (aleph` A) ~< (aleph` A))
34	23, 30, 33	syl11anc 524	. . . . . . . . . . . 12 |- (((A e. On /\ (y e. On /\ y ~<_ (aleph` A))) /\ ((z e. On /\ (aleph` A) ~< z) /\ z C_ y)) -> (aleph` A) ~< (aleph` A))
35	34	expr 418	. . . . . . . . . . 11 |- (((A e. On /\ (y e. On /\ y ~<_ (aleph` A))) /\ (z e. On /\ (aleph` A) ~< z)) -> (z C_ y -> (aleph` A) ~< (aleph` A)))
36	22, 35	mtoi 122	. . . . . . . . . 10 |- (((A e. On /\ (y e. On /\ y ~<_ (aleph` A))) /\ (z e. On /\ (aleph` A) ~< z)) -> -. z C_ y)
37		ordtri1 3693	. . . . . . . . . . . . . . 15 |- ((Ord z /\ Ord y) -> (z C_ y <-> -. y e. z))
38		eloni 3667	. . . . . . . . . . . . . . 15 |- (z e. On -> Ord z)
39		eloni 3667	. . . . . . . . . . . . . . 15 |- (y e. On -> Ord y)
40	37, 38, 39	syl2an 503	. . . . . . . . . . . . . 14 |- ((z e. On /\ y e. On) -> (z C_ y <-> -. y e. z))
41	40	ancoms 484	. . . . . . . . . . . . 13 |- ((y e. On /\ z e. On) -> (z C_ y <-> -. y e. z))
42	41	con2bid 585	. . . . . . . . . . . 12 |- ((y e. On /\ z e. On) -> (y e. z <-> -. z C_ y))
43	42	adantlr 429	. . . . . . . . . . 11 |- (((y e. On /\ y ~<_ (aleph` A)) /\ z e. On) -> (y e. z <-> -. z C_ y))
44	43	ad2ant2lr 446	. . . . . . . . . 10 |- (((A e. On /\ (y e. On /\ y ~<_ (aleph` A))) /\ (z e. On /\ (aleph` A) ~< z)) -> (y e. z <-> -. z C_ y))
45	36, 44	mpbird 213	. . . . . . . . 9 |- (((A e. On /\ (y e. On /\ y ~<_ (aleph` A))) /\ (z e. On /\ (aleph` A) ~< z)) -> y e. z)
46	45	exp31 407	. . . . . . . 8 |- (A e. On -> ((y e. On /\ y ~<_ (aleph` A)) -> ((z e. On /\ (aleph` A) ~< z) -> y e. z)))
47		breq2 3342	. . . . . . . . 9 |- (x = z -> ((aleph` A) ~< x <-> (aleph` A) ~< z))
48	47	elrab 2414	. . . . . . . 8 |- (z e. {x e. On | (aleph` A) ~< x} <-> (z e. On /\ (aleph` A) ~< z))
49	46, 48	syl7ib 233	. . . . . . 7 |- (A e. On -> ((y e. On /\ y ~<_ (aleph` A)) -> (z e. {x e. On | (aleph` A) ~< x} -> y e. z)))
50	49	19.21adv 1666	. . . . . 6 |- (A e. On -> ((y e. On /\ y ~<_ (aleph` A)) -> A.z(z e. {x e. On | (aleph` A) ~< x} -> y e. z)))
51		visset 2295	. . . . . . 7 |- y e. _V
52	51	elint 3220	. . . . . 6 |- (y e. |^|{x e. On | (aleph` A) ~< x} <-> A.z(z e. {x e. On | (aleph` A) ~< x} -> y e. z))
53	50, 52	syl6ibr 230	. . . . 5 |- (A e. On -> ((y e. On /\ y ~<_ (aleph` A)) -> y e. |^|{x e. On | (aleph` A) ~< x}))
54	21, 53	impbid 574	. . . 4 |- (A e. On -> (y e. |^|{x e. On | (aleph` A) ~< x} <-> (y e. On /\ y ~<_ (aleph` A))))
55		breq1 3341	. . . . 5 |- (x = y -> (x ~<_ (aleph` A) <-> y ~<_ (aleph` A)))
56	55	elrab 2414	. . . 4 |- (y e. {x e. On | x ~<_ (aleph` A)} <-> (y e. On /\ y ~<_ (aleph` A)))
57	54, 56	syl6bbr 597	. . 3 |- (A e. On -> (y e. |^|{x e. On | (aleph` A) ~< x} <-> y e. {x e. On | x ~<_ (aleph` A)}))
58	57	eqrdv 1882	. 2 |- (A e. On -> |^|{x e. On | (aleph` A) ~< x} = {x e. On | x ~<_ (aleph` A)})
59	1, 58	eqtrd 1925	1 |- (A e. On -> (aleph` suc A) = {x e. On | x ~<_ (aleph` A)})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  |^|cint 3214   class class class wbr 3338  Ord word 3656  Oncon0 3657  suc csuc 3659  ` cfv 3998   ~<_ cdom 5424   ~< csdm 5425  alephcale 5860

This theorem is referenced by:  omsublim 5887  infenomsub 5889  omsublimOLD 15396  infenomsubOLD 15398

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

Copyright terms: Public domain