Theorem alephcard 8893

Description: Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
alephcard	|- ( card ` ( aleph ` A ) ) = ( aleph ` A )

Proof of Theorem alephcard

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( x = (/) -> ( aleph ` x ) = ( aleph ` (/) ) )
2	1	fveq2d 6195	. . . 4 |- ( x = (/) -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` (/) ) ) )
3	2, 1	eqeq12d 2637	. . 3 |- ( x = (/) -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) ) ) )
4		fveq2 6191	. . . . 5 |- ( x = y -> ( aleph ` x ) = ( aleph ` y ) )
5	4	fveq2d 6195	. . . 4 |- ( x = y -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` y ) ) )
6	5, 4	eqeq12d 2637	. . 3 |- ( x = y -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) )
7		fveq2 6191	. . . . 5 |- ( x = suc y -> ( aleph ` x ) = ( aleph ` suc y ) )
8	7	fveq2d 6195	. . . 4 |- ( x = suc y -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` suc y ) ) )
9	8, 7	eqeq12d 2637	. . 3 |- ( x = suc y -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) ) )
10		fveq2 6191	. . . . 5 |- ( x = A -> ( aleph ` x ) = ( aleph ` A ) )
11	10	fveq2d 6195	. . . 4 |- ( x = A -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` A ) ) )
12	11, 10	eqeq12d 2637	. . 3 |- ( x = A -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` A ) ) = ( aleph ` A ) ) )
13		cardom 8812	. . . 4 |- ( card ` om ) = om
14		aleph0 8889	. . . . 5 |- ( aleph ` (/) ) = om
15	14	fveq2i 6194	. . . 4 |- ( card ` ( aleph ` (/) ) ) = ( card ` om )
16	13, 15, 14	3eqtr4i 2654	. . 3 |- ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) )
17		harcard 8804	. . . . 5 |- ( card ` (har ` ( aleph ` y ) ) ) = (har ` ( aleph ` y ) )
18		alephsuc 8891	. . . . . 6 |- ( y e. On -> ( aleph ` suc y ) = (har ` ( aleph ` y ) ) )
19	18	fveq2d 6195	. . . . 5 |- ( y e. On -> ( card ` ( aleph ` suc y ) ) = ( card ` (har ` ( aleph ` y ) ) ) )
20	17, 19, 18	3eqtr4a 2682	. . . 4 |- ( y e. On -> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) )
21	20	a1d 25	. . 3 |- ( y e. On -> ( ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) ) )
22		vex 3203	. . . . . . 7 |- x e. _V
23		cardiun 8808	. . . . . . 7 |- ( x e. _V -> ( A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` U_ y e. x ( aleph ` y ) ) = U_ y e. x ( aleph ` y ) ) )
24	22, 23	ax-mp 5	. . . . . 6 |- ( A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` U_ y e. x ( aleph ` y ) ) = U_ y e. x ( aleph ` y ) )
25	24	adantl 482	. . . . 5 |- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( card ` U_ y e. x ( aleph ` y ) ) = U_ y e. x ( aleph ` y ) )
26		alephlim 8890	. . . . . . . 8 |- ( ( x e. _V /\ Lim x ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
27	22, 26	mpan 706	. . . . . . 7 |- ( Lim x -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
28	27	adantr 481	. . . . . 6 |- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
29	28	fveq2d 6195	. . . . 5 |- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( card ` ( aleph ` x ) ) = ( card ` U_ y e. x ( aleph ` y ) ) )
30	25, 29, 28	3eqtr4d 2666	. . . 4 |- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( card ` ( aleph ` x ) ) = ( aleph ` x ) )
31	30	ex 450	. . 3 |- ( Lim x -> ( A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` ( aleph ` x ) ) = ( aleph ` x ) ) )
32	3, 6, 9, 12, 16, 21, 31	tfinds 7059	. 2 |- ( A e. On -> ( card ` ( aleph ` A ) ) = ( aleph ` A ) )
33		card0 8784	. . 3 |- ( card ` (/) ) = (/)
34		alephfnon 8888	. . . . . . 7 |- aleph Fn On
35		fndm 5990	. . . . . . 7 |- ( aleph Fn On -> dom aleph = On )
36	34, 35	ax-mp 5	. . . . . 6 |- dom aleph = On
37	36	eleq2i 2693	. . . . 5 |- ( A e. dom aleph <-> A e. On )
38		ndmfv 6218	. . . . 5 |- ( -. A e. dom aleph -> ( aleph ` A ) = (/) )
39	37, 38	sylnbir 321	. . . 4 |- ( -. A e. On -> ( aleph ` A ) = (/) )
40	39	fveq2d 6195	. . 3 |- ( -. A e. On -> ( card ` ( aleph ` A ) ) = ( card ` (/) ) )
41	33, 40, 39	3eqtr4a 2682	. 2 |- ( -. A e. On -> ( card ` ( aleph ` A ) ) = ( aleph ` A ) )
42	32, 41	pm2.61i 176	1 |- ( card ` ( aleph ` A ) ) = ( aleph ` A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   U_ciun 4520   dom cdm 5114   Oncon0 5723   Lim wlim 5724   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065  harchar 8461   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-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:  alephnbtwn2  8895  alephord2  8899  alephsuc2  8903  alephislim  8906  alephsdom  8909  cardaleph  8912  cardalephex  8913  alephval3  8933  alephval2  9394  alephsuc3  9402  alephreg  9404  pwcfsdom  9405