Theorem alephordi 4854

Description: Strict ordering property of the aleph function.

Assertion

Ref	Expression
alephordi	|- (B e. On -> (A e. B -> (aleph` A) ~< (aleph` B)))

Proof of Theorem alephordi

Step	Hyp	Ref	Expression
1		eleq2 1532	. . 3 |- (x = (/) -> (A e. x <-> A e. (/)))
2		fveq2 3715	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
3	2	breq2d 2625	. . 3 |- (x = (/) -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` (/))))
4	1, 3	imbi12d 625	. 2 |- (x = (/) -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. (/) -> (aleph` A) ~< (aleph` (/)))))
5		eleq2 1532	. . 3 |- (x = y -> (A e. x <-> A e. y))
6		fveq2 3715	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
7	6	breq2d 2625	. . 3 |- (x = y -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` y)))
8	5, 7	imbi12d 625	. 2 |- (x = y -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. y -> (aleph` A) ~< (aleph` y))))
9		eleq2 1532	. . 3 |- (x = suc y -> (A e. x <-> A e. suc y))
10		fveq2 3715	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
11	10	breq2d 2625	. . 3 |- (x = suc y -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` suc y)))
12	9, 11	imbi12d 625	. 2 |- (x = suc y -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. suc y -> (aleph` A) ~< (aleph` suc y))))
13		eleq2 1532	. . 3 |- (x = B -> (A e. x <-> A e. B))
14		fveq2 3715	. . . 4 |- (x = B -> (aleph` x) = (aleph` B))
15	14	breq2d 2625	. . 3 |- (x = B -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` B)))
16	13, 15	imbi12d 625	. 2 |- (x = B -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. B -> (aleph` A) ~< (aleph` B))))
17		noel 2280	. . 3 |- -. A e. (/)
18	17	pm2.21i 77	. 2 |- (A e. (/) -> (aleph` A) ~< (aleph` (/)))
19		sdomtr 4460	. . . . . . . . 9 |- (((aleph` A) ~< (aleph` y) /\ (aleph` y) ~< (aleph` suc y)) -> (aleph` A) ~< (aleph` suc y))
20		alephordlem1 4852	. . . . . . . . 9 |- (y e. On -> (aleph` y) ~< (aleph` suc y))
21	19, 20	sylan2 451	. . . . . . . 8 |- (((aleph` A) ~< (aleph` y) /\ y e. On) -> (aleph` A) ~< (aleph` suc y))
22	21	expcom 374	. . . . . . 7 |- (y e. On -> ((aleph` A) ~< (aleph` y) -> (aleph` A) ~< (aleph` suc y)))
23	22	imim2d 25	. . . . . 6 |- (y e. On -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (A e. y -> (aleph` A) ~< (aleph` suc y))))
24	23	com23 32	. . . . 5 |- (y e. On -> (A e. y -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (aleph` A) ~< (aleph` suc y))))
25		fveq2 3715	. . . . . . . . 9 |- (A = y -> (aleph` A) = (aleph` y))
26	25	breq1d 2624	. . . . . . . 8 |- (A = y -> ((aleph` A) ~< (aleph` suc y) <-> (aleph` y) ~< (aleph` suc y)))
27	26, 20	syl5bir 210	. . . . . . 7 |- (A = y -> (y e. On -> (aleph` A) ~< (aleph` suc y)))
28	27	a1d 12	. . . . . 6 |- (A = y -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (y e. On -> (aleph` A) ~< (aleph` suc y))))
29	28	com3r 35	. . . . 5 |- (y e. On -> (A = y -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (aleph` A) ~< (aleph` suc y))))
30	24, 29	jaod 424	. . . 4 |- (y e. On -> ((A e. y \/ A = y) -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (aleph` A) ~< (aleph` suc y))))
31		visset 1809	. . . . 5 |- y e. V
32	31	elsuc2 3034	. . . 4 |- (A e. suc y <-> (A e. y \/ A = y))
33	30, 32	syl5ib 206	. . 3 |- (y e. On -> (A e. suc y -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (aleph` A) ~< (aleph` suc y))))
34	33	com23 32	. 2 |- (y e. On -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (A e. suc y -> (aleph` A) ~< (aleph` suc y))))
35		visset 1809	. . . . . . . . 9 |- x e. V
36		alephlim 4844	. . . . . . . . 9 |- ((x e. V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
37	35, 36	mpan 694	. . . . . . . 8 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
38	37	sseq2d 2085	. . . . . . 7 |- (Lim x -> ((aleph` A) (_ (aleph` x) <-> (aleph` A) (_ U_y e. x (aleph` y)))
39		fveq2 3715	. . . . . . . 8 |- (y = A -> (aleph` y) = (aleph` A))
40	39	ssiun2s 2589	. . . . . . 7 |- (A e. x -> (aleph` A) (_ U_y e. x (aleph` y))
41	38, 40	syl5bir 210	. . . . . 6 |- (Lim x -> (A e. x -> (aleph` A) (_ (aleph` x)))
42		alephon 4845	. . . . . . 7 |- (aleph` A) e. On
43		ssdomg 4395	. . . . . . 7 |- ((aleph` A) e. On -> ((aleph` A) (_ (aleph` x) -> (aleph` A) ~<_ (aleph` x)))
44	42, 43	ax-mp 7	. . . . . 6 |- ((aleph` A) (_ (aleph` x) -> (aleph` A) ~<_ (aleph` x))
45	41, 44	syl6 22	. . . . 5 |- (Lim x -> (A e. x -> (aleph` A) ~<_ (aleph` x)))
46		limsuc 3115	. . . . . . . . . 10 |- (Lim x -> (A e. x <-> suc A e. x))
47		alephordlem2 4853	. . . . . . . . . . 11 |- ((x e. V /\ Lim x) -> (suc A e. x -> (aleph` suc A) ~<_ (aleph` x)))
48	35, 47	mpan 694	. . . . . . . . . 10 |- (Lim x -> (suc A e. x -> (aleph` suc A) ~<_ (aleph` x)))
49	46, 48	sylbid 203	. . . . . . . . 9 |- (Lim x -> (A e. x -> (aleph` suc A) ~<_ (aleph` x)))
50	49	imp 350	. . . . . . . 8 |- ((Lim x /\ A e. x) -> (aleph` suc A) ~<_ (aleph` x))
51		domnsym 4449	. . . . . . . 8 |- ((aleph` suc A) ~<_ (aleph` x) -> -. (aleph` x) ~< (aleph` suc A))
52	50, 51	syl 10	. . . . . . 7 |- ((Lim x /\ A e. x) -> -. (aleph` x) ~< (aleph` suc A))
53		fvex 3723	. . . . . . . . . . 11 |- (aleph` x) e. V
54	53	ensym 4399	. . . . . . . . . 10 |- ((aleph` A) ~~ (aleph` x) -> (aleph` x) ~~ (aleph` A))
55		ensdomtr 4457	. . . . . . . . . . 11 |- (((aleph` x) ~~ (aleph` A) /\ (aleph` A) ~< (aleph` suc A)) -> (aleph` x) ~< (aleph` suc A))
56	55	ex 373	. . . . . . . . . 10 |- ((aleph` x) ~~ (aleph` A) -> ((aleph` A) ~< (aleph` suc A) -> (aleph` x) ~< (aleph` suc A)))
57	54, 56	syl 10	. . . . . . . . 9 |- ((aleph` A) ~~ (aleph` x) -> ((aleph` A) ~< (aleph` suc A) -> (aleph` x) ~< (aleph` suc A)))
58		alephordlem1 4852	. . . . . . . . 9 |- (A e. On -> (aleph` A) ~< (aleph` suc A))
59	57, 58	syl5 21	. . . . . . . 8 |- ((aleph` A) ~~ (aleph` x) -> (A e. On -> (aleph` x) ~< (aleph` suc A)))
60		onelon 2967	. . . . . . . . 9 |- ((x e. On /\ A e. x) -> A e. On)
61		limelon 3027	. . . . . . . . . 10 |- ((x e. V /\ Lim x) -> x e. On)
62	35, 61	mpan 694	. . . . . . . . 9 |- (Lim x -> x e. On)
63	60, 62	sylan 448	. . . . . . . 8 |- ((Lim x /\ A e. x) -> A e. On)
64	59, 63	syl5com 52	. . . . . . 7 |- ((Lim x /\ A e. x) -> ((aleph` A) ~~ (aleph` x) -> (aleph` x) ~< (aleph` suc A)))
65	52, 64	mtod 108	. . . . . 6 |- ((Lim x /\ A e. x) -> -. (aleph` A) ~~ (aleph` x))
66	65	ex 373	. . . . 5 |- (Lim x -> (A e. x -> -. (aleph` A) ~~ (aleph` x)))
67	45, 66	jcad 599	. . . 4 |- (Lim x -> (A e. x -> ((aleph` A) ~<_ (aleph` x) /\ -. (aleph` A) ~~ (aleph` x))))
68		brsdom 4369	. . . 4 |- ((aleph` A) ~< (aleph` x) <-> ((aleph` A) ~<_ (aleph` x) /\ -. (aleph` A) ~~ (aleph` x)))
69	67, 68	syl6ibr 213	. . 3 |- (Lim x -> (A e. x -> (aleph` A) ~< (aleph` x)))
70	69	a1d 12	. 2 |- (Lim x -> (A.y e. x (A e. y