Theorem alephreg 9404

Description: A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)

Assertion

Ref	Expression
alephreg	|- ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A )

Proof of Theorem alephreg

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

Step	Hyp	Ref	Expression
1		alephordilem1 8896	. . . 4 |- ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) )
2		alephon 8892	. . . . . . . . 9 |- ( aleph ` suc A ) e. On
3		cff1 9080	. . . . . . . . 9 |- ( ( aleph ` suc A ) e. On -> E. f ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) )
4	2, 3	ax-mp 5	. . . . . . . 8 |- E. f ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) )
5		fvex 6201	. . . . . . . . . . . . 13 |- ( cf ` ( aleph ` suc A ) ) e. _V
6		fvex 6201	. . . . . . . . . . . . . 14 |- ( f ` y ) e. _V
7	6	sucex 7011	. . . . . . . . . . . . 13 |- suc ( f ` y ) e. _V
8	5, 7	iunex 7147	. . . . . . . . . . . 12 |- U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) e. _V
9		f1f 6101	. . . . . . . . . . . . . 14 |- ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) -> f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) )
10	9	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) )
11		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) )
12	2	oneli 5835	. . . . . . . . . . . . . . . . 17 |- ( x e. ( aleph ` suc A ) -> x e. On )
13		ffvelrn 6357	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) -> ( f ` y ) e. ( aleph ` suc A ) )
14		onelon 5748	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( aleph ` suc A ) e. On /\ ( f ` y ) e. ( aleph ` suc A ) ) -> ( f ` y ) e. On )
15	2, 13, 14	sylancr 695	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) -> ( f ` y ) e. On )
16		onsssuc 5813	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. On /\ ( f ` y ) e. On ) -> ( x C_ ( f ` y ) <-> x e. suc ( f ` y ) ) )
17	15, 16	sylan2 491	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. On /\ ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) ) -> ( x C_ ( f ` y ) <-> x e. suc ( f ` y ) ) )
18	17	anassrs 680	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( x e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) -> ( x C_ ( f ` y ) <-> x e. suc ( f ` y ) ) )
19	18	rexbidva 3049	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> ( E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> E. y e. ( cf ` ( aleph ` suc A ) ) x e. suc ( f ` y ) ) )
20		eliun 4524	. . . . . . . . . . . . . . . . . . 19 |- ( x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) <-> E. y e. ( cf ` ( aleph ` suc A ) ) x e. suc ( f ` y ) )
21	19, 20	syl6bbr 278	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> ( E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) )
22	21	ancoms 469	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ x e. On ) -> ( E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) )
23	12, 22	sylan2 491	. . . . . . . . . . . . . . . 16 |- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ x e. ( aleph ` suc A ) ) -> ( E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) )
24	23	ralbidva 2985	. . . . . . . . . . . . . . 15 |- ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) -> ( A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> A. x e. ( aleph ` suc A ) x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) )
25		dfss3 3592	. . . . . . . . . . . . . . 15 |- ( ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) <-> A. x e. ( aleph ` suc A ) x e. U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) )
26	24, 25	syl6bbr 278	. . . . . . . . . . . . . 14 |- ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) -> ( A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) <-> ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) )
27	26	biimpa 501	. . . . . . . . . . . . 13 |- ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) -> ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) )
28	10, 11, 27	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) )
29		ssdomg 8001	. . . . . . . . . . . 12 |- ( U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) e. _V -> ( ( aleph ` suc A ) C_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) -> ( aleph ` suc A ) ~<_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ) )
30	8, 28, 29	mpsyl 68	. . . . . . . . . . 11 |- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> ( aleph ` suc A ) ~<_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) )
31		simprl 794	. . . . . . . . . . . 12 |- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> A e. On )
32		suceloni 7013	. . . . . . . . . . . . . . . . . 18 |- ( A e. On -> suc A e. On )
33		alephislim 8906	. . . . . . . . . . . . . . . . . . 19 |- ( suc A e. On <-> Lim ( aleph ` suc A ) )
34		limsuc 7049	. . . . . . . . . . . . . . . . . . 19 |- ( Lim ( aleph ` suc A ) -> ( ( f ` y ) e. ( aleph ` suc A ) <-> suc ( f ` y ) e. ( aleph ` suc A ) ) )
35	33, 34	sylbi 207	. . . . . . . . . . . . . . . . . 18 |- ( suc A e. On -> ( ( f ` y ) e. ( aleph ` suc A ) <-> suc ( f ` y ) e. ( aleph ` suc A ) ) )
36	32, 35	syl 17	. . . . . . . . . . . . . . . . 17 |- ( A e. On -> ( ( f ` y ) e. ( aleph ` suc A ) <-> suc ( f ` y ) e. ( aleph ` suc A ) ) )
37		breq1 4656	. . . . . . . . . . . . . . . . . . 19 |- ( z = suc ( f ` y ) -> ( z ~< ( aleph ` suc A ) <-> suc ( f ` y ) ~< ( aleph ` suc A ) ) )
38		alephcard 8893	. . . . . . . . . . . . . . . . . . . 20 |- ( card ` ( aleph ` suc A ) ) = ( aleph ` suc A )
39		iscard 8801	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( card ` ( aleph ` suc A ) ) = ( aleph ` suc A ) <-> ( ( aleph ` suc A ) e. On /\ A. z e. ( aleph ` suc A ) z ~< ( aleph ` suc A ) ) )
40	39	simprbi 480	. . . . . . . . . . . . . . . . . . . 20 |- ( ( card ` ( aleph ` suc A ) ) = ( aleph ` suc A ) -> A. z e. ( aleph ` suc A ) z ~< ( aleph ` suc A ) )
41	38, 40	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- A. z e. ( aleph ` suc A ) z ~< ( aleph ` suc A )
42	37, 41	vtoclri 3283	. . . . . . . . . . . . . . . . . 18 |- ( suc ( f ` y ) e. ( aleph ` suc A ) -> suc ( f ` y ) ~< ( aleph ` suc A ) )
43		alephsucdom 8902	. . . . . . . . . . . . . . . . . 18 |- ( A e. On -> ( suc ( f ` y ) ~<_ ( aleph ` A ) <-> suc ( f ` y ) ~< ( aleph ` suc A ) ) )
44	42, 43	syl5ibr 236	. . . . . . . . . . . . . . . . 17 |- ( A e. On -> ( suc ( f ` y ) e. ( aleph ` suc A ) -> suc ( f ` y ) ~<_ ( aleph ` A ) ) )
45	36, 44	sylbid 230	. . . . . . . . . . . . . . . 16 |- ( A e. On -> ( ( f ` y ) e. ( aleph ` suc A ) -> suc ( f ` y ) ~<_ ( aleph ` A ) ) )
46	13, 45	syl5 34	. . . . . . . . . . . . . . 15 |- ( A e. On -> ( ( f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) /\ y e. ( cf ` ( aleph ` suc A ) ) ) -> suc ( f ` y ) ~<_ ( aleph ` A ) ) )
47	46	expdimp 453	. . . . . . . . . . . . . 14 |- ( ( A e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> ( y e. ( cf ` ( aleph ` suc A ) ) -> suc ( f ` y ) ~<_ ( aleph ` A ) ) )
48	47	ralrimiv 2965	. . . . . . . . . . . . 13 |- ( ( A e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> A. y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( aleph ` A ) )
49		iundom 9364	. . . . . . . . . . . . 13 |- ( ( ( cf ` ( aleph ` suc A ) ) e. _V /\ A. y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( aleph ` A ) ) -> U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) )
50	5, 48, 49	sylancr 695	. . . . . . . . . . . 12 |- ( ( A e. On /\ f : ( cf ` ( aleph ` suc A ) ) --> ( aleph ` suc A ) ) -> U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) )
51	31, 10, 50	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) )
52		domtr 8009	. . . . . . . . . . 11 |- ( ( ( aleph ` suc A ) ~<_ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) /\ U_ y e. ( cf ` ( aleph ` suc A ) ) suc ( f ` y ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) )
53	30, 51, 52	syl2anc 693	. . . . . . . . . 10 |- ( ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) /\ ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) )
54	53	expcom 451	. . . . . . . . 9 |- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) )
55	54	exlimdv 1861	. . . . . . . 8 |- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( E. f ( f : ( cf ` ( aleph ` suc A ) ) -1-1-> ( aleph ` suc A ) /\ A. x e. ( aleph ` suc A ) E. y e. ( cf ` ( aleph ` suc A ) ) x C_ ( f ` y ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ) )
56	4, 55	mpi 20	. . . . . . 7 |- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) )
57		alephgeom 8905	. . . . . . . . . 10 |- ( A e. On <-> om C_ ( aleph ` A ) )
58		alephon 8892	. . . . . . . . . . 11 |- ( aleph ` A ) e. On
59		infxpen 8837	. . . . . . . . . . 11 |- ( ( ( aleph ` A ) e. On /\ om C_ ( aleph ` A ) ) -> ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) )
60	58, 59	mpan 706	. . . . . . . . . 10 |- ( om C_ ( aleph ` A ) -> ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) )
61	57, 60	sylbi 207	. . . . . . . . 9 |- ( A e. On -> ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) )
62		breq1 4656	. . . . . . . . . . . 12 |- ( z = ( cf ` ( aleph ` suc A ) ) -> ( z ~< ( aleph ` suc A ) <-> ( cf ` ( aleph ` suc A ) ) ~< ( aleph ` suc A ) ) )
63	62, 41	vtoclri 3283	. . . . . . . . . . 11 |- ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( cf ` ( aleph ` suc A ) ) ~< ( aleph ` suc A ) )
64		alephsucdom 8902	. . . . . . . . . . 11 |- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) ~<_ ( aleph ` A ) <-> ( cf ` ( aleph ` suc A ) ) ~< ( aleph ` suc A ) ) )
65	63, 64	syl5ibr 236	. . . . . . . . . 10 |- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( cf ` ( aleph ` suc A ) ) ~<_ ( aleph ` A ) ) )
66		fvex 6201	. . . . . . . . . . 11 |- ( aleph ` A ) e. _V
67	66	xpdom1 8059	. . . . . . . . . 10 |- ( ( cf ` ( aleph ` suc A ) ) ~<_ ( aleph ` A ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( ( aleph ` A ) X. ( aleph ` A ) ) )
68	65, 67	syl6 35	. . . . . . . . 9 |- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( ( aleph ` A ) X. ( aleph ` A ) ) ) )
69		domentr 8015	. . . . . . . . . 10 |- ( ( ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( ( aleph ` A ) X. ( aleph ` A ) ) /\ ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) )
70	69	expcom 451	. . . . . . . . 9 |- ( ( ( aleph ` A ) X. ( aleph ` A ) ) ~~ ( aleph ` A ) -> ( ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( ( aleph ` A ) X. ( aleph ` A ) ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) ) )
71	61, 68, 70	sylsyld 61	. . . . . . . 8 |- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) ) )
72	71	imp 445	. . . . . . 7 |- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) )
73		domtr 8009	. . . . . . 7 |- ( ( ( aleph ` suc A ) ~<_ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) /\ ( ( cf ` ( aleph ` suc A ) ) X. ( aleph ` A ) ) ~<_ ( aleph ` A ) ) -> ( aleph ` suc A ) ~<_ ( aleph ` A ) )
74	56, 72, 73	syl2anc 693	. . . . . 6 |- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> ( aleph ` suc A ) ~<_ ( aleph ` A ) )
75		domnsym 8086	. . . . . 6 |- ( ( aleph ` suc A ) ~<_ ( aleph ` A ) -> -. ( aleph ` A ) ~< ( aleph ` suc A ) )
76	74, 75	syl 17	. . . . 5 |- ( ( A e. On /\ ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) ) -> -. ( aleph ` A ) ~< ( aleph ` suc A ) )
77	76	ex 450	. . . 4 |- ( A e. On -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> -. ( aleph ` A ) ~< ( aleph ` suc A ) ) )
78	1, 77	mt2d 131	. . 3 |- ( A e. On -> -. ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) )
79		cfon 9077	. . . . 5 |- ( cf ` ( aleph ` suc A ) ) e. On
80		cfle 9076	. . . . . 6 |- ( cf ` ( aleph ` suc A ) ) C_ ( aleph ` suc A )
81		onsseleq 5765	. . . . . 6 |- ( ( ( cf ` ( aleph ` suc A ) ) e. On /\ ( aleph ` suc A ) e. On ) -> ( ( cf ` ( aleph ` suc A ) ) C_ ( aleph ` suc A ) <-> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) \/ ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) ) ) )
82	80, 81	mpbii 223	. . . . 5 |- ( ( ( cf ` ( aleph ` suc A ) ) e. On /\ ( aleph ` suc A ) e. On ) -> ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) \/ ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) ) )
83	79, 2, 82	mp2an 708	. . . 4 |- ( ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) \/ ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) )
84	83	ori 390	. . 3 |- ( -. ( cf ` ( aleph ` suc A ) ) e. ( aleph ` suc A ) -> ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) )
85	78, 84	syl 17	. 2 |- ( A e. On -> ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) )
86		cf0 9073	. . 3 |- ( cf ` (/) ) = (/)
87		alephfnon 8888	. . . . . . . 8 |- aleph Fn On
88		fndm 5990	. . . . . . . 8 |- ( aleph Fn On -> dom aleph = On )
89	87, 88	ax-mp 5	. . . . . . 7 |- dom aleph = On
90	89	eleq2i 2693	. . . . . 6 |- ( suc A e. dom aleph <-> suc A e. On )
91		sucelon 7017	. . . . . 6 |- ( A e. On <-> suc A e. On )
92	90, 91	bitr4i 267	. . . . 5 |- ( suc A e. dom aleph <-> A e. On )
93		ndmfv 6218	. . . . 5 |- ( -. suc A e. dom aleph -> ( aleph ` suc A ) = (/) )
94	92, 93	sylnbir 321	. . . 4 |- ( -. A e. On -> ( aleph ` suc A ) = (/) )
95	94	fveq2d 6195	. . 3 |- ( -. A e. On -> ( cf ` ( aleph ` suc A ) ) = ( cf ` (/) ) )
96	86, 95, 94	3eqtr4a 2682	. 2 |- ( -. A e. On -> ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A ) )
97	85, 96	pm2.61i 176	1 |- ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U_ciun 4520   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Oncon0 5723   Lim wlim 5724   suc csuc 5725   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   ` cfv 5888   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761   alephcale 8762   cfccf 8763

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  ax-ac2 9285

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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766  df-cf 8767  df-acn 8768  df-ac 8939

This theorem is referenced by:  pwcfsdom  9405