Theorem alephcard 8344

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 5792	. . . . 5 |- ( x = (/) -> ( aleph ` x ) = ( aleph ` (/) ) )
2	1	fveq2d 5796	. . . 4 |- ( x = (/) -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` (/) ) ) )
3	2, 1	eqeq12d 2473	. . 3 |- ( x = (/) -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) ) ) )
4		fveq2 5792	. . . . 5 |- ( x = y -> ( aleph ` x ) = ( aleph ` y ) )
5	4	fveq2d 5796	. . . 4 |- ( x = y -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` y ) ) )
6	5, 4	eqeq12d 2473	. . 3 |- ( x = y -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) )
7		fveq2 5792	. . . . 5 |- ( x = suc y -> ( aleph ` x ) = ( aleph ` suc y ) )
8	7	fveq2d 5796	. . . 4 |- ( x = suc y -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` suc y ) ) )
9	8, 7	eqeq12d 2473	. . 3 |- ( x = suc y -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) ) )
10		fveq2 5792	. . . . 5 |- ( x = A -> ( aleph ` x ) = ( aleph ` A ) )
11	10	fveq2d 5796	. . . 4 |- ( x = A -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` A ) ) )
12	11, 10	eqeq12d 2473	. . 3 |- ( x = A -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` A ) ) = ( aleph ` A ) ) )
13		cardom 8260	. . . 4 |- ( card ` om ) = om
14		aleph0 8340	. . . . 5 |- ( aleph ` (/) ) = om
15	14	fveq2i 5795	. . . 4 |- ( card ` ( aleph ` (/) ) ) = ( card ` om )
16	13, 15, 14	3eqtr4i 2490	. . 3 |- ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) )
17		harcard 8252	. . . . 5 |- ( card ` (har ` ( aleph ` y ) ) ) = (har ` ( aleph ` y ) )
18		alephsuc 8342	. . . . . 6 |- ( y e. On -> ( aleph ` suc y ) = (har ` ( aleph ` y ) ) )
19	18	fveq2d 5796	. . . . 5 |- ( y e. On -> ( card ` ( aleph ` suc y ) ) = ( card ` (har ` ( aleph ` y ) ) ) )
20	17, 19, 18	3eqtr4a 2518	. . . 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 3074	. . . . . . 7 |- x e. _V
23		cardiun 8256	. . . . . . 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 466	. . . . 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 8341	. . . . . . . 8 |- ( ( x e. _V /\ Lim x ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
27	22, 26	mpan 670	. . . . . . 7 |- ( Lim x -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
28	27	adantr 465	. . . . . 6 |- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) )
29	28	fveq2d 5796	. . . . 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 2502	. . . 4 |- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( card ` ( aleph ` x ) ) = ( aleph ` x ) )
31	30	ex 434	. . 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 6573	. 2 |- ( A e. On -> ( card ` ( aleph ` A ) ) = ( aleph ` A ) )
33		card0 8232	. . 3 |- ( card ` (/) ) = (/)
34		alephfnon 8339	. . . . . . 7 |- aleph Fn On
35		fndm 5611	. . . . . . 7 |- ( aleph Fn On -> dom aleph = On )
36	34, 35	ax-mp 5	. . . . . 6 |- dom aleph = On
37	36	eleq2i 2529	. . . . 5 |- ( A e. dom aleph <-> A e. On )
38		ndmfv 5816	. . . . 5 |- ( -. A e. dom aleph -> ( aleph ` A ) = (/) )
39	37, 38	sylnbir 307	. . . 4 |- ( -. A e. On -> ( aleph ` A ) = (/) )
40	39	fveq2d 5796	. . 3 |- ( -. A e. On -> ( card ` ( aleph ` A ) ) = ( card ` (/) ) )
41	33, 40, 39	3eqtr4a 2518	. 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 369   = wceq 1370   e. wcel 1758   A.wral 2795   _Vcvv 3071   (/)c0 3738   U_ciun 4272   Oncon0 4820   Lim wlim 4821   suc csuc 4822   dom cdm 4941   Fn wfn 5514   ` cfv 5519   omcom 6579  harchar 7875   cardccrd 8209   alephcale 8210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-om 6580  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-oi 7828  df-har 7877  df-card 8213  df-aleph 8214

This theorem is referenced by:  alephnbtwn2  8346  alephord2  8350  alephsuc2  8354  alephislim  8357  alephsdom  8360  cardaleph  8363  cardalephex  8364  alephval3  8384  alephval2  8840  alephsuc3  8848  alephreg  8850  pwcfsdom  8851