Theorem alephcard 8442

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 5848	. . . . 5 |- ( x = (/) -> ( aleph ` x ) = ( aleph ` (/) ) )
2	1	fveq2d 5852	. . . 4 |- ( x = (/) -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` (/) ) ) )
3	2, 1	eqeq12d 2476	. . 3 |- ( x = (/) -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) ) ) )
4		fveq2 5848	. . . . 5 |- ( x = y -> ( aleph ` x ) = ( aleph ` y ) )
5	4	fveq2d 5852	. . . 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 5848	. . . . 5 |- ( x = suc y -> ( aleph ` x ) = ( aleph ` suc y ) )
8	7	fveq2d 5852	. . . 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 5848	. . . . 5 |- ( x = A -> ( aleph ` x ) = ( aleph ` A ) )
11	10	fveq2d 5852	. . . 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 8358	. . . 4 |- ( card ` om ) = om
14		aleph0 8438	. . . . 5 |- ( aleph ` (/) ) = om
15	14	fveq2i 5851	. . . 4 |- ( card ` ( aleph ` (/) ) ) = ( card ` om )
16	13, 15, 14	3eqtr4i 2493	. . 3 |- ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) )
17		harcard 8350	. . . . 5 |- ( card ` (har ` ( aleph ` y ) ) ) = (har ` ( aleph ` y ) )
18		alephsuc 8440	. . . . . 6 |- ( y e. On -> ( aleph ` suc y ) = (har ` ( aleph ` y ) ) )
19	18	fveq2d 5852	. . . . 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 25	. . 3 |- ( y e. On -> ( ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) ) )
22		vex 3109	. . . . . . 7 |- x e. _V
23		cardiun 8354	. . . . . . 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 464	. . . . 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 8439	. . . . . . . 8 |- ( ( x e. _V /\ Lim x ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
27	22, 26	mpan 668	. . . . . . 7 |- ( Lim x -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
28	27	adantr 463	. . . . . 6 |- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
29	28	fveq2d 5852	. . . . 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 432	. . 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 6667	. 2 |- ( A e. On -> ( card ` ( aleph ` A ) ) = ( aleph ` A ) )
33		card0 8330	. . 3 |- ( card ` (/) ) = (/)
34		alephfnon 8437	. . . . . . 7 |- aleph Fn On
35		fndm 5662	. . . . . . 7 |- ( aleph Fn On -> dom aleph = On )
36	34, 35	ax-mp 5	. . . . . 6 |- dom aleph = On
37	36	eleq2i 2532	. . . . 5 |- ( A e. dom aleph <-> A e. On )
38		ndmfv 5872	. . . . 5 |- ( -. A e. dom aleph -> ( aleph ` A ) = (/) )
39	37, 38	sylnbir 305	. . . 4 |- ( -. A e. On -> ( aleph ` A ) = (/) )
40	39	fveq2d 5852	. . 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 164	1 |- ( card ` ( aleph ` A ) ) = ( aleph ` A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   _Vcvv 3106   (/)c0 3783   U_ciun 4315   Oncon0 4867   Lim wlim 4868   suc csuc 4869   dom cdm 4988   Fn wfn 5565   ` cfv 5570   omcom 6673  harchar 7974   cardccrd 8307   alephcale 8308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-har 7976  df-card 8311  df-aleph 8312

This theorem is referenced by:  alephnbtwn2  8444  alephord2  8448  alephsuc2  8452  alephislim  8455  alephsdom  8458  cardaleph  8461  cardalephex  8462  alephval3  8482  alephval2  8938  alephsuc3  8946  alephreg  8948  pwcfsdom  8949