Theorem alephordi 6022

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 1958	. . 3 |- (x = (/) -> (A e. x <-> A e. (/)))
2		fveq2 4681	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
3	2	breq2d 3350	. . 3 |- (x = (/) -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` (/))))
4	1, 3	imbi12d 688	. 2 |- (x = (/) -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. (/) -> (aleph` A) ~< (aleph` (/)))))
5		eleq2 1958	. . 3 |- (x = y -> (A e. x <-> A e. y))
6		fveq2 4681	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
7	6	breq2d 3350	. . 3 |- (x = y -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` y)))
8	5, 7	imbi12d 688	. 2 |- (x = y -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. y -> (aleph` A) ~< (aleph` y))))
9		eleq2 1958	. . 3 |- (x = suc y -> (A e. x <-> A e. suc y))
10		fveq2 4681	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
11	10	breq2d 3350	. . 3 |- (x = suc y -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` suc y)))
12	9, 11	imbi12d 688	. 2 |- (x = suc y -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. suc y -> (aleph` A) ~< (aleph` suc y))))
13		eleq2 1958	. . 3 |- (x = B -> (A e. x <-> A e. B))
14		fveq2 4681	. . . 4 |- (x = B -> (aleph` x) = (aleph` B))
15	14	breq2d 3350	. . 3 |- (x = B -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` B)))
16	13, 15	imbi12d 688	. 2 |- (x = B -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. B -> (aleph` A) ~< (aleph` B))))
17		noel 2879	. . 3 |- -. A e. (/)
18	17	pm2.21i 93	. 2 |- (A e. (/) -> (aleph` A) ~< (aleph` (/)))
19		sdomtr 5537	. . . . . . . . 9 |- (((aleph` A) ~< (aleph` y) /\ (aleph` y) ~< (aleph` suc y)) -> (aleph` A) ~< (aleph` suc y))
20		alephordlem1 6020	. . . . . . . . 9 |- (y e. On -> (aleph` y) ~< (aleph` suc y))
21	19, 20	sylan2 500	. . . . . . . 8 |- (((aleph` A) ~< (aleph` y) /\ y e. On) -> (aleph` A) ~< (aleph` suc y))
22	21	expcom 403	. . . . . . 7 |- (y e. On -> ((aleph` A) ~< (aleph` y) -> (aleph` A) ~< (aleph` suc y)))
23	22	imim2d 28	. . . . . 6 |- (y e. On -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (A e. y -> (aleph` A) ~< (aleph` suc y))))
24	23	com23 36	. . . . 5 |- (y e. On -> (A e. y -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (aleph` A) ~< (aleph` suc y))))
25		fveq2 4681	. . . . . . . . 9 |- (A = y -> (aleph` A) = (aleph` y))
26	25	breq1d 3348	. . . . . . . 8 |- (A = y -> ((aleph` A) ~< (aleph` suc y) <-> (aleph` y) ~< (aleph` suc y)))
27	26, 20	syl5bir 227	. . . . . . 7 |- (A = y -> (y e. On -> (aleph` A) ~< (aleph` suc y)))
28	27	a1d 15	. . . . . 6 |- (A = y -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (y e. On -> (aleph` A) ~< (aleph` suc y))))
29	28	com3r 39	. . . . 5 |- (y e. On -> (A = y -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (aleph` A) ~< (aleph` suc y))))
30	24, 29	jaod 469	. . . 4 |- (y e. On -> ((A e. y \/ A = y) -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (aleph` A) ~< (aleph` suc y))))
31		visset 2295	. . . . 5 |- y e. _V
32	31	elsuc2 3735	. . . 4 |- (A e. suc y <-> (A e. y \/ A = y))
33	30, 32	syl5ib 223	. . 3 |- (y e. On -> (A e. suc y -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (aleph` A) ~< (aleph` suc y))))
34	33	com23 36	. 2 |- (y e. On -> ((A e. y -> (aleph` A) ~< (aleph` y)) -> (A e. suc y -> (aleph` A) ~< (aleph` suc y))))
35		visset 2295	. . . . . . . . 9 |- x e. _V
36		alephlim 5875	. . . . . . . . 9 |- ((x e. _V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
37	35, 36	mpan 759	. . . . . . . 8 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
38	37	sseq2d 2645	. . . . . . 7 |- (Lim x -> ((aleph` A) C_ (aleph` x) <-> (aleph` A) C_ U_y e. x (aleph` y)))
39		fveq2 4681	. . . . . . . 8 |- (y = A -> (aleph` y) = (aleph` A))
40	39	ssiun2s 3297	. . . . . . 7 |- (A e. x -> (aleph` A) C_ U_y e. x (aleph` y))
41	38, 40	syl5bir 227	. . . . . 6 |- (Lim x -> (A e. x -> (aleph` A) C_ (aleph` x)))
42		alephon 5876	. . . . . . 7 |- (aleph` A) e. On
43		ssdomg 5467	. . . . . . 7 |- ((aleph` A) e. On -> ((aleph` A) C_ (aleph` x) -> (aleph` A) ~<_ (aleph` x)))
44	42, 43	ax-mp 7	. . . . . 6 |- ((aleph` A) C_ (aleph` x) -> (aleph` A) ~<_ (aleph` x))
45	41, 44	syl6 25	. . . . 5 |- (Lim x -> (A e. x -> (aleph` A) ~<_ (aleph` x)))
46		limsuc 3933	. . . . . . . . . 10 |- (Lim x -> (A e. x <-> suc A e. x))
47		alephordlem2 6021	. . . . . . . . . . 11 |- ((x e. _V /\ Lim x) -> (suc A e. x -> (aleph` suc A) ~<_ (aleph` x)))
48	35, 47	mpan 759	. . . . . . . . . 10 |- (Lim x -> (suc A e. x -> (aleph` suc A) ~<_ (aleph` x)))
49	46, 48	sylbid 220	. . . . . . . . 9 |- (Lim x -> (A e. x -> (aleph` suc A) ~<_ (aleph` x)))
50	49	imp 377	. . . . . . . 8 |- ((Lim x /\ A e. x) -> (aleph` suc A) ~<_ (aleph` x))
51		domnsym 5526	. . . . . . . 8 |- ((aleph` suc A) ~<_ (aleph` x) -> -. (aleph` x) ~< (aleph` suc A))
52	50, 51	syl 12	. . . . . . 7 |- ((Lim x /\ A e. x) -> -. (aleph` x) ~< (aleph` suc A))
53		fvex 4689	. . . . . . . . . . 11 |- (aleph` x) e. _V
54	53	ensym 5471	. . . . . . . . . 10 |- ((aleph` A) ~~ (aleph` x) -> (aleph` x) ~~ (aleph` A))
55		ensdomtr 5534	. . . . . . . . . . 11 |- (((aleph` x) ~~ (aleph` A) /\ (aleph` A) ~< (aleph` suc A)) -> (aleph` x) ~< (aleph` suc A))
56	55	ex 402	. . . . . . . . . 10 |- ((aleph` x) ~~ (aleph` A) -> ((aleph` A) ~< (aleph` suc A) -> (aleph` x) ~< (aleph` suc A)))
57	54, 56	syl 12	. . . . . . . . 9 |- ((aleph` A) ~~ (aleph` x) -> ((aleph` A) ~< (aleph` suc A) -> (aleph` x) ~< (aleph` suc A)))
58		alephordlem1 6020	. . . . . . . . 9 |- (A e. On -> (aleph` A) ~< (aleph` suc A))
59	57, 58	syl5 20	. . . . . . . 8 |- ((aleph` A) ~~ (aleph` x) -> (A e. On -> (aleph` x) ~< (aleph` suc A)))
60		onelon 3683	. . . . . . . . 9 |- ((x e. On /\ A e. x) -> A e. On)
61		limelon 3727	. . . . . . . . . 10 |- ((x e. _V /\ Lim x) -> x e. On)
62	35, 61	mpan 759	. . . . . . . . 9 |- (Lim x -> x e. On)
63	60, 62	sylan 497	. . . . . . . 8 |- ((Lim x /\ A e. x) -> A e. On)
64	59, 63	syl5com 63	. . . . . . 7 |- ((Lim x /\ A e. x) -> ((aleph` A) ~~ (aleph` x) -> (aleph` x) ~< (aleph` suc A)))
65	52, 64	mtod 123	. . . . . 6 |- ((Lim x /\ A e. x) -> -. (aleph` A) ~~ (aleph` x))
66	65	ex 402	. . . . 5 |- (Lim x -> (A e. x -> -. (aleph` A) ~~ (aleph` x)))
67	45, 66	jcad 661	. . . 4 |- (Lim x -> (A e. x -> ((aleph` A) ~<_ (aleph` x) /\ -. (aleph` A) ~~ (aleph` x))))
68		brsdom 5440	. . . 4 |- ((aleph` A) ~< (aleph` x) <-> ((aleph` A) ~<_ (aleph` x) /\ -. (aleph` A) ~~ (aleph` x)))
69	67, 68	syl6ibr 230	. . 3 |- (Lim x -> (A e. x -> (aleph` A) ~< (aleph` x)))
70	69	a1d 15	. 2 |- (Lim x -> (A.y e. x (A e. y -> (aleph` A) ~< (aleph` y)) -> (A e. x -> (aleph` A) ~< (aleph` x))))
71	4, 8, 12, 16, 18, 34, 70	tfinds 3942	1 |- (B e. On -> (A e. B -> (aleph` A) ~< (aleph` B)))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  (/)c0 2875  U_ciun 3255   class class class wbr 3338  Oncon0 3657  Lim wlim 3658  suc csuc 3659  ` cfv 3998   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425  alephcale 5860

This theorem is referenced by:  alephord 6023  alephval2 6050

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-aleph 5863

Copyright terms: Public domain