Theorem alephsuc3 8744

Description: An alternate representation of a successor aleph. Compare alephsuc 8238 and alephsuc2 8250. Equality can be obtained by taking the card of the right-hand side then using alephcard 8240 and carden 8715. (Contributed by NM, 23-Oct-2004.)

Assertion

Ref	Expression
alephsuc3	|- ( A e. On -> ( aleph ` suc A ) ~~ { x e. On | x ~~ ( aleph ` A ) } )

Distinct variable group:   x, A

Proof of Theorem alephsuc3

Step	Hyp	Ref	Expression
1		alephsuc2 8250	. . . . 5 |- ( A e. On -> ( aleph ` suc A ) = { x e. On | x ~<_ ( aleph ` A ) } )
2		alephcard 8240	. . . . . . 7 |- ( card ` ( aleph ` A ) ) = ( aleph ` A )
3		alephon 8239	. . . . . . . . 9 |- ( aleph ` A ) e. On
4		onenon 8119	. . . . . . . . 9 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
5	3, 4	ax-mp 5	. . . . . . . 8 |- ( aleph ` A ) e. dom card
6		cardval2 8161	. . . . . . . 8 |- ( ( aleph ` A ) e. dom card -> ( card ` ( aleph ` A ) ) = { x e. On | x ~< ( aleph ` A ) } )
7	5, 6	ax-mp 5	. . . . . . 7 |- ( card ` ( aleph ` A ) ) = { x e. On | x ~< ( aleph ` A ) }
8	2, 7	eqtr3i 2465	. . . . . 6 |- ( aleph ` A ) = { x e. On | x ~< ( aleph ` A ) }
9	8	a1i 11	. . . . 5 |- ( A e. On -> ( aleph ` A ) = { x e. On | x ~< ( aleph ` A ) } )
10	1, 9	difeq12d 3475	. . . 4 |- ( A e. On -> ( ( aleph ` suc A ) \ ( aleph ` A ) ) = ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) )
11		difrab 3624	. . . . 5 |- ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) = { x e. On | ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) }
12		bren2 7340	. . . . . . 7 |- ( x ~~ ( aleph ` A ) <-> ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) )
13	12	a1i 11	. . . . . 6 |- ( x e. On -> ( x ~~ ( aleph ` A ) <-> ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) ) )
14	13	rabbiia 2961	. . . . 5 |- { x e. On | x ~~ ( aleph ` A ) } = { x e. On | ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) }
15	11, 14	eqtr4i 2466	. . . 4 |- ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) = { x e. On | x ~~ ( aleph ` A ) }
16	10, 15	syl6req 2492	. . 3 |- ( A e. On -> { x e. On | x ~~ ( aleph ` A ) } = ( ( aleph ` suc A ) \ ( aleph ` A ) ) )
17		alephon 8239	. . . . 5 |- ( aleph ` suc A ) e. On
18		onenon 8119	. . . . 5 |- ( ( aleph ` suc A ) e. On -> ( aleph ` suc A ) e. dom card )
19	17, 18	mp1i 12	. . . 4 |- ( A e. On -> ( aleph ` suc A ) e. dom card )
20		sucelon 6428	. . . . . 6 |- ( A e. On <-> suc A e. On )
21		alephgeom 8252	. . . . . 6 |- ( suc A e. On <-> om C_ ( aleph ` suc A ) )
22	20, 21	bitri 249	. . . . 5 |- ( A e. On <-> om C_ ( aleph ` suc A ) )
23		fvex 5701	. . . . . 6 |- ( aleph ` suc A ) e. _V
24		ssdomg 7355	. . . . . 6 |- ( ( aleph ` suc A ) e. _V -> ( om C_ ( aleph ` suc A ) -> om ~<_ ( aleph ` suc A ) ) )
25	23, 24	ax-mp 5	. . . . 5 |- ( om C_ ( aleph ` suc A ) -> om ~<_ ( aleph ` suc A ) )
26	22, 25	sylbi 195	. . . 4 |- ( A e. On -> om ~<_ ( aleph ` suc A ) )
27		alephordilem1 8243	. . . 4 |- ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) )
28		infdif 8378	. . . 4 |- ( ( ( aleph ` suc A ) e. dom card /\ om ~<_ ( aleph ` suc A ) /\ ( aleph ` A ) ~< ( aleph ` suc A ) ) -> ( ( aleph ` suc A ) \ ( aleph ` A ) ) ~~ ( aleph ` suc A ) )
29	19, 26, 27, 28	syl3anc 1218	. . 3 |- ( A e. On -> ( ( aleph ` suc A ) \ ( aleph ` A ) ) ~~ ( aleph ` suc A ) )
30	16, 29	eqbrtrd 4312	. 2 |- ( A e. On -> { x e. On | x ~~ ( aleph ` A ) } ~~ ( aleph ` suc A ) )
31	30	ensymd 7360	1 |- ( A e. On -> ( aleph ` suc A ) ~~ { x e. On | x ~~ ( aleph ` A ) } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   {crab 2719   _Vcvv 2972   \ cdif 3325   C_ wss 3328   class class class wbr 4292   Oncon0 4719   suc csuc 4721   dom cdm 4840   ` cfv 5418   omcom 6476   ~~ cen 7307   ~<_ cdom 7308   ~< csdm 7309   cardccrd 8105   alephcale 8106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-oi 7724  df-har 7773  df-card 8109  df-aleph 8110  df-cda 8337

This theorem is referenced by: (None)