Theorem alephsuc3 8958

Description: An alternate representation of a successor aleph. Compare alephsuc 8452 and alephsuc2 8464. Equality can be obtained by taking the card of the right-hand side then using alephcard 8454 and carden 8929. (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 8464	. . . . 5 |- ( A e. On -> ( aleph ` suc A ) = { x e. On | x ~<_ ( aleph ` A ) } )
2		alephcard 8454	. . . . . . 7 |- ( card ` ( aleph ` A ) ) = ( aleph ` A )
3		alephon 8453	. . . . . . . . 9 |- ( aleph ` A ) e. On
4		onenon 8333	. . . . . . . . 9 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
5	3, 4	ax-mp 5	. . . . . . . 8 |- ( aleph ` A ) e. dom card
6		cardval2 8375	. . . . . . . 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 2474	. . . . . 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 3608	. . . 4 |- ( A e. On -> ( ( aleph ` suc A ) \ ( aleph ` A ) ) = ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) )
11		difrab 3757	. . . . 5 |- ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) = { x e. On | ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) }
12		bren2 7548	. . . . . . 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 3084	. . . . 5 |- { x e. On | x ~~ ( aleph ` A ) } = { x e. On | ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) }
15	11, 14	eqtr4i 2475	. . . 4 |- ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) = { x e. On | x ~~ ( aleph ` A ) }
16	10, 15	syl6req 2501	. . 3 |- ( A e. On -> { x e. On | x ~~ ( aleph ` A ) } = ( ( aleph ` suc A ) \ ( aleph ` A ) ) )
17		alephon 8453	. . . . 5 |- ( aleph ` suc A ) e. On
18		onenon 8333	. . . . 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 6637	. . . . . 6 |- ( A e. On <-> suc A e. On )
21		alephgeom 8466	. . . . . 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 5866	. . . . . 6 |- ( aleph ` suc A ) e. _V
24		ssdomg 7563	. . . . . 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 8457	. . . 4 |- ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) )
28		infdif 8592	. . . 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 1229	. . 3 |- ( A e. On -> ( ( aleph ` suc A ) \ ( aleph ` A ) ) ~~ ( aleph ` suc A ) )
30	16, 29	eqbrtrd 4457	. 2 |- ( A e. On -> { x e. On | x ~~ ( aleph ` A ) } ~~ ( aleph ` suc A ) )
31	30	ensymd 7568	1 |- ( A e. On -> ( aleph ` suc A ) ~~ { x e. On | x ~~ ( aleph ` A ) } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   {crab 2797   _Vcvv 3095   \ cdif 3458   C_ wss 3461   class class class wbr 4437   Oncon0 4868   suc csuc 4870   dom cdm 4989   ` cfv 5578   omcom 6685   ~~ cen 7515   ~<_ cdom 7516   ~< csdm 7517   cardccrd 8319   alephcale 8320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-oi 7938  df-har 7987  df-card 8323  df-aleph 8324  df-cda 8551

This theorem is referenced by: (None)