Theorem alephsuc3 9005

Description: An alternate representation of a successor aleph. Compare alephsuc 8499 and alephsuc2 8511. Equality can be obtained by taking the card of the right-hand side then using alephcard 8501 and carden 8976. (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 8511	. . . . 5 |- ( A e. On -> ( aleph ` suc A ) = { x e. On | x ~<_ ( aleph ` A ) } )
2		alephcard 8501	. . . . . . 7 |- ( card ` ( aleph ` A ) ) = ( aleph ` A )
3		alephon 8500	. . . . . . . . 9 |- ( aleph ` A ) e. On
4		onenon 8383	. . . . . . . . 9 |- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
5	3, 4	ax-mp 5	. . . . . . . 8 |- ( aleph ` A ) e. dom card
6		cardval2 8425	. . . . . . . 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 2475	. . . . . 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 3552	. . . 4 |- ( A e. On -> ( ( aleph ` suc A ) \ ( aleph ` A ) ) = ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) )
11		difrab 3717	. . . . 5 |- ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) = { x e. On | ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) }
12		bren2 7600	. . . . . . 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 3033	. . . . 5 |- { x e. On | x ~~ ( aleph ` A ) } = { x e. On | ( x ~<_ ( aleph ` A ) /\ -. x ~< ( aleph ` A ) ) }
15	11, 14	eqtr4i 2476	. . . 4 |- ( { x e. On | x ~<_ ( aleph ` A ) } \ { x e. On | x ~< ( aleph ` A ) } ) = { x e. On | x ~~ ( aleph ` A ) }
16	10, 15	syl6req 2502	. . 3 |- ( A e. On -> { x e. On | x ~~ ( aleph ` A ) } = ( ( aleph ` suc A ) \ ( aleph ` A ) ) )
17		alephon 8500	. . . . 5 |- ( aleph ` suc A ) e. On
18		onenon 8383	. . . . 5 |- ( ( aleph ` suc A ) e. On -> ( aleph ` suc A ) e. dom card )
19	17, 18	mp1i 13	. . . 4 |- ( A e. On -> ( aleph ` suc A ) e. dom card )
20		sucelon 6644	. . . . . 6 |- ( A e. On <-> suc A e. On )
21		alephgeom 8513	. . . . . 6 |- ( suc A e. On <-> om C_ ( aleph ` suc A ) )
22	20, 21	bitri 253	. . . . 5 |- ( A e. On <-> om C_ ( aleph ` suc A ) )
23		fvex 5875	. . . . . 6 |- ( aleph ` suc A ) e. _V
24		ssdomg 7615	. . . . . 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 199	. . . 4 |- ( A e. On -> om ~<_ ( aleph ` suc A ) )
27		alephordilem1 8504	. . . 4 |- ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) )
28		infdif 8639	. . . 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 1268	. . 3 |- ( A e. On -> ( ( aleph ` suc A ) \ ( aleph ` A ) ) ~~ ( aleph ` suc A ) )
30	16, 29	eqbrtrd 4423	. 2 |- ( A e. On -> { x e. On | x ~~ ( aleph ` A ) } ~~ ( aleph ` suc A ) )
31	30	ensymd 7620	1 |- ( A e. On -> ( aleph ` suc A ) ~~ { x e. On | x ~~ ( aleph ` A ) } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   {crab 2741   _Vcvv 3045   \ cdif 3401   C_ wss 3404   class class class wbr 4402   dom cdm 4834   Oncon0 5423   suc csuc 5425   ` cfv 5582   omcom 6692   ~~ cen 7566   ~<_ cdom 7567   ~< csdm 7568   cardccrd 8369   alephcale 8370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-oi 8025  df-har 8073  df-card 8373  df-aleph 8374  df-cda 8598

This theorem is referenced by: (None)