Theorem alephcard 8526

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 5887	. . . . 5 |- ( x = (/) -> ( aleph ` x ) = ( aleph ` (/) ) )
2	1	fveq2d 5891	. . . 4 |- ( x = (/) -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` (/) ) ) )
3	2, 1	eqeq12d 2476	. . 3 |- ( x = (/) -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) ) ) )
4		fveq2 5887	. . . . 5 |- ( x = y -> ( aleph ` x ) = ( aleph ` y ) )
5	4	fveq2d 5891	. . . 4 |- ( x = y -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` y ) ) )
6	5, 4	eqeq12d 2476	. . 3 |- ( x = y -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) )
7		fveq2 5887	. . . . 5 |- ( x = suc y -> ( aleph ` x ) = ( aleph ` suc y ) )
8	7	fveq2d 5891	. . . 4 |- ( x = suc y -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` suc y ) ) )
9	8, 7	eqeq12d 2476	. . 3 |- ( x = suc y -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) ) )
10		fveq2 5887	. . . . 5 |- ( x = A -> ( aleph ` x ) = ( aleph ` A ) )
11	10	fveq2d 5891	. . . 4 |- ( x = A -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` A ) ) )
12	11, 10	eqeq12d 2476	. . 3 |- ( x = A -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` A ) ) = ( aleph ` A ) ) )
13		cardom 8445	. . . 4 |- ( card ` om ) = om
14		aleph0 8522	. . . . 5 |- ( aleph ` (/) ) = om
15	14	fveq2i 5890	. . . 4 |- ( card ` ( aleph ` (/) ) ) = ( card ` om )
16	13, 15, 14	3eqtr4i 2493	. . 3 |- ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) )
17		harcard 8437	. . . . 5 |- ( card ` (har ` ( aleph ` y ) ) ) = (har ` ( aleph ` y ) )
18		alephsuc 8524	. . . . . 6 |- ( y e. On -> ( aleph ` suc y ) = (har ` ( aleph ` y ) ) )
19	18	fveq2d 5891	. . . . 5 |- ( y e. On -> ( card ` ( aleph ` suc y ) ) = ( card ` (har ` ( aleph ` y ) ) ) )
20	17, 19, 18	3eqtr4a 2521	. . . 4 |- ( y e. On -> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) )
21	20	a1d 26	. . 3 |- ( y e. On -> ( ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) ) )
22		vex 3059	. . . . . . 7 |- x e. _V
23		cardiun 8441	. . . . . . 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 472	. . . . 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 8523	. . . . . . . 8 |- ( ( x e. _V /\ Lim x ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
27	22, 26	mpan 681	. . . . . . 7 |- ( Lim x -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
28	27	adantr 471	. . . . . 6 |- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
29	28	fveq2d 5891	. . . . 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 2505	. . . 4 |- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( card ` ( aleph ` x ) ) = ( aleph ` x ) )
31	30	ex 440	. . 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 6712	. 2 |- ( A e. On -> ( card ` ( aleph ` A ) ) = ( aleph ` A ) )
33		card0 8417	. . 3 |- ( card ` (/) ) = (/)
34		alephfnon 8521	. . . . . . 7 |- aleph Fn On
35		fndm 5696	. . . . . . 7 |- ( aleph Fn On -> dom aleph = On )
36	34, 35	ax-mp 5	. . . . . 6 |- dom aleph = On
37	36	eleq2i 2531	. . . . 5 |- ( A e. dom aleph <-> A e. On )
38		ndmfv 5911	. . . . 5 |- ( -. A e. dom aleph -> ( aleph ` A ) = (/) )
39	37, 38	sylnbir 313	. . . 4 |- ( -. A e. On -> ( aleph ` A ) = (/) )
40	39	fveq2d 5891	. . 3 |- ( -. A e. On -> ( card ` ( aleph ` A ) ) = ( card ` (/) ) )
41	33, 40, 39	3eqtr4a 2521	. 2 |- ( -. A e. On -> ( card ` ( aleph ` A ) ) = ( aleph ` A ) )
42	32, 41	pm2.61i 169	1 |- ( card ` ( aleph ` A ) ) = ( aleph ` A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 375   = wceq 1454   e. wcel 1897   A.wral 2748   _Vcvv 3056   (/)c0 3742   U_ciun 4291   dom cdm 4852   Oncon0 5441   Lim wlim 5442   suc csuc 5443   Fn wfn 5595   ` cfv 5600   omcom 6718  harchar 8096   cardccrd 8394   alephcale 8395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-oi 8050  df-har 8098  df-card 8398  df-aleph 8399

This theorem is referenced by:  alephnbtwn2  8528  alephord2  8532  alephsuc2  8536  alephislim  8539  alephsdom  8542  cardaleph  8545  cardalephex  8546  alephval3  8566  alephval2  9022  alephsuc3  9030  alephreg  9032  pwcfsdom  9033