Theorem alephval3 8933

Description: An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.)

Assertion

Ref	Expression
alephval3	|- ( A e. On -> ( aleph ` A ) = |^| { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } )

Distinct variable group:   x, y, A

Proof of Theorem alephval3

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephcard 8893	. . . 4 |- ( card ` ( aleph ` A ) ) = ( aleph ` A )
2	1	a1i 11	. . 3 |- ( A e. On -> ( card ` ( aleph ` A ) ) = ( aleph ` A ) )
3		alephgeom 8905	. . . 4 |- ( A e. On <-> om C_ ( aleph ` A ) )
4	3	biimpi 206	. . 3 |- ( A e. On -> om C_ ( aleph ` A ) )
5		alephord2i 8900	. . . . 5 |- ( A e. On -> ( y e. A -> ( aleph ` y ) e. ( aleph ` A ) ) )
6		elirr 8505	. . . . . . 7 |- -. ( aleph ` y ) e. ( aleph ` y )
7		eleq2 2690	. . . . . . 7 |- ( ( aleph ` A ) = ( aleph ` y ) -> ( ( aleph ` y ) e. ( aleph ` A ) <-> ( aleph ` y ) e. ( aleph ` y ) ) )
8	6, 7	mtbiri 317	. . . . . 6 |- ( ( aleph ` A ) = ( aleph ` y ) -> -. ( aleph ` y ) e. ( aleph ` A ) )
9	8	con2i 134	. . . . 5 |- ( ( aleph ` y ) e. ( aleph ` A ) -> -. ( aleph ` A ) = ( aleph ` y ) )
10	5, 9	syl6 35	. . . 4 |- ( A e. On -> ( y e. A -> -. ( aleph ` A ) = ( aleph ` y ) ) )
11	10	ralrimiv 2965	. . 3 |- ( A e. On -> A. y e. A -. ( aleph ` A ) = ( aleph ` y ) )
12		fvex 6201	. . . 4 |- ( aleph ` A ) e. _V
13		fveq2 6191	. . . . . 6 |- ( x = ( aleph ` A ) -> ( card ` x ) = ( card ` ( aleph ` A ) ) )
14		id 22	. . . . . 6 |- ( x = ( aleph ` A ) -> x = ( aleph ` A ) )
15	13, 14	eqeq12d 2637	. . . . 5 |- ( x = ( aleph ` A ) -> ( ( card ` x ) = x <-> ( card ` ( aleph ` A ) ) = ( aleph ` A ) ) )
16		sseq2 3627	. . . . 5 |- ( x = ( aleph ` A ) -> ( om C_ x <-> om C_ ( aleph ` A ) ) )
17		eqeq1 2626	. . . . . . 7 |- ( x = ( aleph ` A ) -> ( x = ( aleph ` y ) <-> ( aleph ` A ) = ( aleph ` y ) ) )
18	17	notbid 308	. . . . . 6 |- ( x = ( aleph ` A ) -> ( -. x = ( aleph ` y ) <-> -. ( aleph ` A ) = ( aleph ` y ) ) )
19	18	ralbidv 2986	. . . . 5 |- ( x = ( aleph ` A ) -> ( A. y e. A -. x = ( aleph ` y ) <-> A. y e. A -. ( aleph ` A ) = ( aleph ` y ) ) )
20	15, 16, 19	3anbi123d 1399	. . . 4 |- ( x = ( aleph ` A ) -> ( ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) <-> ( ( card ` ( aleph ` A ) ) = ( aleph ` A ) /\ om C_ ( aleph ` A ) /\ A. y e. A -. ( aleph ` A ) = ( aleph ` y ) ) ) )
21	12, 20	elab 3350	. . 3 |- ( ( aleph ` A ) e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } <-> ( ( card ` ( aleph ` A ) ) = ( aleph ` A ) /\ om C_ ( aleph ` A ) /\ A. y e. A -. ( aleph ` A ) = ( aleph ` y ) ) )
22	2, 4, 11, 21	syl3anbrc 1246	. 2 |- ( A e. On -> ( aleph ` A ) e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } )
23		cardalephex 8913	. . . . . . . . . 10 |- ( om C_ z -> ( ( card ` z ) = z <-> E. y e. On z = ( aleph ` y ) ) )
24	23	biimpac 503	. . . . . . . . 9 |- ( ( ( card ` z ) = z /\ om C_ z ) -> E. y e. On z = ( aleph ` y ) )
25		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( z = ( aleph ` y ) -> ( z e. ( aleph ` A ) <-> ( aleph ` y ) e. ( aleph ` A ) ) )
26		alephord2 8899	. . . . . . . . . . . . . . . . 17 |- ( ( y e. On /\ A e. On ) -> ( y e. A <-> ( aleph ` y ) e. ( aleph ` A ) ) )
27	26	bicomd 213	. . . . . . . . . . . . . . . 16 |- ( ( y e. On /\ A e. On ) -> ( ( aleph ` y ) e. ( aleph ` A ) <-> y e. A ) )
28	25, 27	sylan9bbr 737	. . . . . . . . . . . . . . 15 |- ( ( ( y e. On /\ A e. On ) /\ z = ( aleph ` y ) ) -> ( z e. ( aleph ` A ) <-> y e. A ) )
29	28	biimpcd 239	. . . . . . . . . . . . . 14 |- ( z e. ( aleph ` A ) -> ( ( ( y e. On /\ A e. On ) /\ z = ( aleph ` y ) ) -> y e. A ) )
30		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( y e. On /\ A e. On ) /\ z = ( aleph ` y ) ) -> z = ( aleph ` y ) )
31	30	a1i 11	. . . . . . . . . . . . . 14 |- ( z e. ( aleph ` A ) -> ( ( ( y e. On /\ A e. On ) /\ z = ( aleph ` y ) ) -> z = ( aleph ` y ) ) )
32	29, 31	jcad 555	. . . . . . . . . . . . 13 |- ( z e. ( aleph ` A ) -> ( ( ( y e. On /\ A e. On ) /\ z = ( aleph ` y ) ) -> ( y e. A /\ z = ( aleph ` y ) ) ) )
33	32	exp4c 636	. . . . . . . . . . . 12 |- ( z e. ( aleph ` A ) -> ( y e. On -> ( A e. On -> ( z = ( aleph ` y ) -> ( y e. A /\ z = ( aleph ` y ) ) ) ) ) )
34	33	com3r 87	. . . . . . . . . . 11 |- ( A e. On -> ( z e. ( aleph ` A ) -> ( y e. On -> ( z = ( aleph ` y ) -> ( y e. A /\ z = ( aleph ` y ) ) ) ) ) )
35	34	imp4b 613	. . . . . . . . . 10 |- ( ( A e. On /\ z e. ( aleph ` A ) ) -> ( ( y e. On /\ z = ( aleph ` y ) ) -> ( y e. A /\ z = ( aleph ` y ) ) ) )
36	35	reximdv2 3014	. . . . . . . . 9 |- ( ( A e. On /\ z e. ( aleph ` A ) ) -> ( E. y e. On z = ( aleph ` y ) -> E. y e. A z = ( aleph ` y ) ) )
37	24, 36	syl5 34	. . . . . . . 8 |- ( ( A e. On /\ z e. ( aleph ` A ) ) -> ( ( ( card ` z ) = z /\ om C_ z ) -> E. y e. A z = ( aleph ` y ) ) )
38	37	imp 445	. . . . . . 7 |- ( ( ( A e. On /\ z e. ( aleph ` A ) ) /\ ( ( card ` z ) = z /\ om C_ z ) ) -> E. y e. A z = ( aleph ` y ) )
39		dfrex2 2996	. . . . . . 7 |- ( E. y e. A z = ( aleph ` y ) <-> -. A. y e. A -. z = ( aleph ` y ) )
40	38, 39	sylib 208	. . . . . 6 |- ( ( ( A e. On /\ z e. ( aleph ` A ) ) /\ ( ( card ` z ) = z /\ om C_ z ) ) -> -. A. y e. A -. z = ( aleph ` y ) )
41		nan 604	. . . . . 6 |- ( ( ( A e. On /\ z e. ( aleph ` A ) ) -> -. ( ( ( card ` z ) = z /\ om C_ z ) /\ A. y e. A -. z = ( aleph ` y ) ) ) <-> ( ( ( A e. On /\ z e. ( aleph ` A ) ) /\ ( ( card ` z ) = z /\ om C_ z ) ) -> -. A. y e. A -. z = ( aleph ` y ) ) )
42	40, 41	mpbir 221	. . . . 5 |- ( ( A e. On /\ z e. ( aleph ` A ) ) -> -. ( ( ( card ` z ) = z /\ om C_ z ) /\ A. y e. A -. z = ( aleph ` y ) ) )
43	42	ex 450	. . . 4 |- ( A e. On -> ( z e. ( aleph ` A ) -> -. ( ( ( card ` z ) = z /\ om C_ z ) /\ A. y e. A -. z = ( aleph ` y ) ) ) )
44		vex 3203	. . . . . . 7 |- z e. _V
45		fveq2 6191	. . . . . . . . 9 |- ( x = z -> ( card ` x ) = ( card ` z ) )
46		id 22	. . . . . . . . 9 |- ( x = z -> x = z )
47	45, 46	eqeq12d 2637	. . . . . . . 8 |- ( x = z -> ( ( card ` x ) = x <-> ( card ` z ) = z ) )
48		sseq2 3627	. . . . . . . 8 |- ( x = z -> ( om C_ x <-> om C_ z ) )
49		eqeq1 2626	. . . . . . . . . 10 |- ( x = z -> ( x = ( aleph ` y ) <-> z = ( aleph ` y ) ) )
50	49	notbid 308	. . . . . . . . 9 |- ( x = z -> ( -. x = ( aleph ` y ) <-> -. z = ( aleph ` y ) ) )
51	50	ralbidv 2986	. . . . . . . 8 |- ( x = z -> ( A. y e. A -. x = ( aleph ` y ) <-> A. y e. A -. z = ( aleph ` y ) ) )
52	47, 48, 51	3anbi123d 1399	. . . . . . 7 |- ( x = z -> ( ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) <-> ( ( card ` z ) = z /\ om C_ z /\ A. y e. A -. z = ( aleph ` y ) ) ) )
53	44, 52	elab 3350	. . . . . 6 |- ( z e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } <-> ( ( card ` z ) = z /\ om C_ z /\ A. y e. A -. z = ( aleph ` y ) ) )
54		df-3an 1039	. . . . . 6 |- ( ( ( card ` z ) = z /\ om C_ z /\ A. y e. A -. z = ( aleph ` y ) ) <-> ( ( ( card ` z ) = z /\ om C_ z ) /\ A. y e. A -. z = ( aleph ` y ) ) )
55	53, 54	bitri 264	. . . . 5 |- ( z e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } <-> ( ( ( card ` z ) = z /\ om C_ z ) /\ A. y e. A -. z = ( aleph ` y ) ) )
56	55	notbii 310	. . . 4 |- ( -. z e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } <-> -. ( ( ( card ` z ) = z /\ om C_ z ) /\ A. y e. A -. z = ( aleph ` y ) ) )
57	43, 56	syl6ibr 242	. . 3 |- ( A e. On -> ( z e. ( aleph ` A ) -> -. z e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } ) )
58	57	ralrimiv 2965	. 2 |- ( A e. On -> A. z e. ( aleph ` A ) -. z e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } )
59		cardon 8770	. . . . . 6 |- ( card ` x ) e. On
60		eleq1 2689	. . . . . 6 |- ( ( card ` x ) = x -> ( ( card ` x ) e. On <-> x e. On ) )
61	59, 60	mpbii 223	. . . . 5 |- ( ( card ` x ) = x -> x e. On )
62	61	3ad2ant1 1082	. . . 4 |- ( ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) -> x e. On )
63	62	abssi 3677	. . 3 |- { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } C_ On
64		oneqmini 5776	. . 3 |- ( { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } C_ On -> ( ( ( aleph ` A ) e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } /\ A. z e. ( aleph ` A ) -. z e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } ) -> ( aleph ` A ) = |^| { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } ) )
65	63, 64	ax-mp 5	. 2 |- ( ( ( aleph ` A ) e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } /\ A. z e. ( aleph ` A ) -. z e. { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } ) -> ( aleph ` A ) = |^| { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } )
66	22, 58, 65	syl2anc 693	1 |- ( A e. On -> ( aleph ` A ) = |^| { x | ( ( card ` x ) = x /\ om C_ x /\ A. y e. A -. x = ( aleph ` y ) ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   |^|cint 4475   Oncon0 5723   ` cfv 5888   omcom 7065   cardccrd 8761   alephcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by: (None)