Theorem alephval2 4902

Description: An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229.

Assertion

Ref	Expression
alephval2	|- ((A e. On /\ (/) e. A) -> (aleph` A) = |^|{x e. On | A.y e. A (aleph` y) ~< x})

Distinct variable group:   x,y,A

Proof of Theorem alephval2

Step	Hyp	Ref	Expression
1		ssrab2 2131	. . 3 |- {x e. On | A.y e. A (aleph` y) ~< x} (_ On
2		oneqmini 3017	. . 3 |- ({x e. On | A.y e. A (aleph` y) ~< x} (_ On -> (((aleph` A) e. {x e. On | A.y e. A (aleph` y) ~< x} /\ A.z e. (aleph` A) -. z e. {x e. On | A.y e. A (aleph` y) ~< x}) -> (aleph` A) = |^|{x e. On | A.y e. A (aleph` y) ~< x}))
3	1, 2	ax-mp 7	. 2 |- (((aleph` A) e. {x e. On | A.y e. A (aleph` y) ~< x} /\ A.z e. (aleph` A) -. z e. {x e. On | A.y e. A (aleph` y) ~< x}) -> (aleph` A) = |^|{x e. On | A.y e. A (aleph` y) ~< x})
4		alephordi 4874	. . . . . 6 |- (A e. On -> (y e. A -> (aleph` y) ~< (aleph` A)))
5	4	r19.21aiv 1713	. . . . 5 |- (A e. On -> A.y e. A (aleph` y) ~< (aleph` A))
6		alephon 4865	. . . . 5 |- (aleph` A) e. On
7	5, 6	jctil 292	. . . 4 |- (A e. On -> ((aleph` A) e. On /\ A.y e. A (aleph` y) ~< (aleph` A)))
8		breq2 2623	. . . . . 6 |- (x = (aleph` A) -> ((aleph` y) ~< x <-> (aleph` y) ~< (aleph` A)))
9	8	ralbidv 1663	. . . . 5 |- (x = (aleph` A) -> (A.y e. A (aleph` y) ~< x <-> A.y e. A (aleph` y) ~< (aleph` A)))
10	9	elrab 1905	. . . 4 |- ((aleph` A) e. {x e. On | A.y e. A (aleph` y) ~< x} <-> ((aleph` A) e. On /\ A.y e. A (aleph` y) ~< (aleph` A)))
11	7, 10	sylibr 200	. . 3 |- (A e. On -> (aleph` A) e. {x e. On | A.y e. A (aleph` y) ~< x})
12	11	adantr 389	. 2 |- ((A e. On /\ (/) e. A) -> (aleph` A) e. {x e. On | A.y e. A (aleph` y) ~< x})
13		omex 4627	. . . . . . 7 |- om e. V
14		visset 1813	. . . . . . 7 |- z e. V
15		entri3 4841	. . . . . . 7 |- ((om e. V /\ z e. V) -> (om ~<_ z \/ z ~<_ om))
16	13, 14, 15	mp2an 697	. . . . . 6 |- (om ~<_ z \/ z ~<_ om)
17		alephord 4875	. . . . . . . . . . . . . . 15 |- ((x e. On /\ A e. On) -> (x e. A <-> (aleph` x) ~< (aleph` A)))
18	17	ancoms 436	. . . . . . . . . . . . . 14 |- ((A e. On /\ x e. On) -> (x e. A <-> (aleph` x) ~< (aleph` A)))
19		breq1 2622	. . . . . . . . . . . . . . 15 |- ((card` z) = (aleph` x) -> ((card` z) ~< (aleph` A) <-> (aleph` x) ~< (aleph` A)))
20		cardid 4828	. . . . . . . . . . . . . . . 16 |- (card` z) ~~ z
21		sdomen1 4481	. . . . . . . . . . . . . . . 16 |- ((z e. V /\ (card` z) ~~ z) -> ((card` z) ~< (aleph` A) <-> z ~< (aleph` A)))
22	14, 20, 21	mp2an 697	. . . . . . . . . . . . . . 15 |- ((card` z) ~< (aleph` A) <-> z ~< (aleph` A))
23	19, 22	syl5rbbr 535	. . . . . . . . . . . . . 14 |- ((card` z) = (aleph` x) -> ((aleph` x) ~< (aleph` A) <-> z ~< (aleph` A)))
24	18, 23	sylan9bb 540	. . . . . . . . . . . . 13 |- (((A e. On /\ x e. On) /\ (card` z) = (aleph` x)) -> (x e. A <-> z ~< (aleph` A)))
25		fveq2 3724	. . . . . . . . . . . . . . . . . 18 |- (y = x -> (aleph` y) = (aleph` x))
26	25	breq1d 2629	. . . . . . . . . . . . . . . . 17 |- (y = x -> ((aleph` y) ~< z <-> (aleph` x) ~< z))
27	26	rcla4v 1873	. . . . . . . . . . . . . . . 16 |- (x e. A -> (A.y e. A (aleph` y) ~< z -> (aleph` x) ~< z))
28		sdomirr 4472	. . . . . . . . . . . . . . . . 17 |- -. (aleph` x) ~< (aleph` x)
29		breq2 2623	. . . . . . . . . . . . . . . . . 18 |- ((card` z) = (aleph` x) -> ((aleph` x) ~< (card` z) <-> (aleph` x) ~< (aleph` x)))
30		sdomen2 4482	. . . . . . . . . . . . . . . . . . 19 |- ((z e. V /\ (card` z) ~~ z) -> ((aleph` x) ~< (card` z) <-> (aleph` x) ~< z))
31	14, 20, 30	mp2an 697	. . . . . . . . . . . . . . . . . 18 |- ((aleph` x) ~< (card` z) <-> (aleph` x) ~< z)
32	29, 31	syl5bbr 534	. . . . . . . . . . . . . . . . 17 |- ((card` z) = (aleph` x) -> ((aleph` x) ~< z <-> (aleph` x) ~< (aleph` x)))
33	28, 32	mtbiri 717	. . . . . . . . . . . . . . . 16 |- ((card` z) = (aleph` x) -> -. (aleph` x) ~< z)
34	27, 33	nsyli 121	. . . . . . . . . . . . . . 15 |- (x e. A -> ((card` z) = (aleph` x) -> -. A.y e. A (aleph` y) ~< z))
35	34	com12 11	. . . . . . . . . . . . . 14 |- ((card` z) = (aleph` x) -> (x e. A -> -. A.y e. A (aleph` y) ~< z))
36	35	adantl 388	. . . . . . . . . . . . 13 |- (((A e. On /\ x e. On) /\ (card` z) = (aleph` x)) -> (x e. A -> -. A.y e. A (aleph` y) ~< z))
37	24, 36	sylbird 205	. . . . . . . . . . . 12 |- (((A e. On /\ x e. On) /\ (card` z) = (aleph` x)) -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z))
38	37	exp31 376	. . . . . . . . . . 11 |- (A e. On -> (x e. On -> ((card` z) = (aleph` x) -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z))))
39	38	r19.23adv 1746	. . . . . . . . . 10 |- (A e. On -> (E.x e. On (card` z) = (aleph` x) -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z)))
40		cardidm 4849	. . . . . . . . . . 11 |- (card` (card` z)) = (card` z)
41		cardalephex 4886	. . . . . . . . . . 11 |- (om (_ (card` z) -> ((card` (card` z)) = (card` z) <-> E.x e. On (card` z) = (aleph` x)))
42	40, 41	mpbii 193	. . . . . . . . . 10 |- (om (_ (card` z) -> E.x e. On (card` z) = (aleph` x))
43	39, 42	syl5 21	. . . . . . . . 9 |- (A e. On -> (om (_ (card` z) -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z)))
44		carddom 4836	. . . . . . . . . . 11 |- ((om e. V /\ z e. V) -> ((card` om) (_ (card` z) <-> om ~<_ z))
45	13, 14, 44	mp2an 697	. . . . . . . . . 10 |- ((card` om) (_ (card` z) <-> om ~<_ z)
46		cardom 4825	. . . . . . . . . . 11 |- (card` om) = om
47	46	sseq1i 2085	. . . . . . . . . 10 |- ((card` om) (_ (card` z) <-> om (_ (card` z))
48	45, 47	bitr3 175	. . . . . . . . 9 |- (om ~<_ z <-> om (_ (card` z))
49	43, 48	syl5ib 206	. . . . . . . 8 |- (A e. On -> (om ~<_ z -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z)))
50	49	adantr 389	. . . . . . 7 |- ((A e. On /\ (/) e. A) -> (om ~<_ z -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z)))
51		ne0i 2286	. . . . . . . . . . . . . 14 |- ((/) e. A -> A =/= (/))
52		r19.2z 2347	. . . . . . . . . . . . . . . 16 |- ((A =/= (/) /\ A.y e. A -. (aleph` y) ~< z) -> E.y e. A -. (aleph` y) ~< z)
53	52	ex 373	. . . . . . . . . . . . . . 15 |- (A =/= (/) -> (A.y e. A -. (aleph` y) ~< z -> E.y e. A -. (aleph` y) ~< z))
54		domtr 4415	. . . . . . . . . . . . . . . . . . . . 21 |- ((z ~<_ om /\ om ~<_ (aleph` y)) -> z ~<_ (aleph` y))
55		alephgeom 4882	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. On <-> om (_ (aleph` y))
56		ssdomg 4408	. . . . . . . . . . . . . . . . . . . . . . 23 |- (om e. V -> (om (_ (aleph` y) -> om ~<_ (aleph` y)))
57	13, 56	ax-mp 7	. . . . . . . . . . . . . . . . . . . . . 22 |- (om (_ (aleph` y) -> om ~<_ (aleph` y))
58	55, 57	sylbi 199	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. On -> om ~<_ (aleph` y))
59	54, 58	sylan2 451	. . . . . . . . . . . . . . . . . . . 20 |- ((z ~<_ om /\ y e. On) -> z ~<_ (aleph` y))
60		domnsym 4463	. . . . . . . . . . . . . . . . . . . 20 |- (z ~<_ (aleph` y) -> -. (aleph` y) ~< z)
61	59, 60	syl 10	. . . . . . . . . . . . . . . . . . 19 |- ((z ~<_ om /\ y e. On) -> -. (aleph` y) ~< z)
62		onelon 2972	. . . . . . . . . . . . . . . . . . 19 |- ((A e. On /\ y e. A) -> y e. On)
63	61, 62	sylan2 451	. . . . . . . . . . . . . . . . . 18 |- ((z ~<_ om /\ (A e. On /\ y e. A)) -> -. (aleph` y) ~< z)
64	63	exp32 377	. . . . . . . . . . . . . . . . 17 |- (z ~<_ om -> (A e. On -> (y e. A -> -. (aleph` y) ~< z)))
65	64	imp 350	. . . . . . . . . . . . . . . 16 |- ((z ~<_ om /\ A e. On) -> (y e. A -> -. (aleph` y) ~<