Theorem alephsuc3 8951

Description: An alternate representation of a successor aleph. Compare alephsuc 8445 and alephsuc2 8457. Equality can be obtained by taking the card of the right-hand side then using alephcard 8447 and carden 8922. (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 8457	. . . . 5 |- ( A e. On -> ( aleph ` suc A ) = { x e. On | x ~<_ ( aleph ` A ) } )
2		alephcard 8447	. . . . . . 7 |- ( card ` ( aleph ` A ) ) = ( aleph ` A )
3		alephon 8446	. . . . . . . . 9 |- ( aleph ` A ) e. On
4		onenon 8326	. . . . . . . . 9 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
5	3, 4	ax-mp 5	. . . . . . . 8 |- ( aleph ` A ) e. dom card
6		cardval2 8368	. . . . . . . 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 2498	. . . . . 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 3623	. . . 4 |- ( A e. On -> ( ( aleph ` suc A ) \ ( aleph ` A ) ) = ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) )
11		difrab 3772	. . . . 5 |- ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) = { x e. On | ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) }
12		bren2 7543	. . . . . . 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 3102	. . . . 5 |- { x e. On | x ~~ ( aleph ` A ) } = { x e. On | ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) }
15	11, 14	eqtr4i 2499	. . . 4 |- ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) = { x e. On | x ~~ ( aleph ` A ) }
16	10, 15	syl6req 2525	. . 3 |- ( A e. On -> { x e. On | x ~~ ( aleph ` A ) } = ( ( aleph ` suc A ) \ ( aleph ` A ) ) )
17		alephon 8446	. . . . 5 |- ( aleph ` suc A ) e. On
18		onenon 8326	. . . . 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 6630	. . . . . 6 |- ( A e. On <-> suc A e. On )
21		alephgeom 8459	. . . . . 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 5874	. . . . . 6 |- ( aleph ` suc A ) e. _V
24		ssdomg 7558	. . . . . 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 8450	. . . 4 |- ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) )
28		infdif 8585	. . . 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 1228	. . 3 |- ( A e. On -> ( ( aleph ` suc A ) \ ( aleph ` A ) ) ~~ ( aleph ` suc A ) )
30	16, 29	eqbrtrd 4467	. 2 |- ( A e. On -> { x e. On | x ~~ ( aleph ` A ) } ~~ ( aleph ` suc A ) )
31	30	ensymd 7563	1 |- ( A e. On -> ( aleph ` suc A ) ~~ { x e. On | x ~~ ( aleph ` A ) } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   {crab 2818   _Vcvv 3113   \ cdif 3473   C_ wss 3476   class class class wbr 4447   Oncon0 4878   suc csuc 4880   dom cdm 4999   ` cfv 5586   omcom 6678   ~~ cen 7510   ~<_ cdom 7511   ~< csdm 7512   cardccrd 8312   alephcale 8313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-oi 7931  df-har 7980  df-card 8316  df-aleph 8317  df-cda 8544

This theorem is referenced by: (None)