Theorem alephordi 5024

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 1578	. . 3 |- (x = (/) -> (A e. x <-> A e. (/)))
2		fveq2 3835	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
3	2	breq2d 2703	. . 3 |- (x = (/) -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` (/))))
4	1, 3	imbi12d 629	. 2 |- (x = (/) -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. (/) -> (aleph` A) ~< (aleph` (/)))))
5		eleq2 1578	. . 3 |- (x = y -> (A e. x <-> A e. y))
6		fveq2 3835	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
7	6	breq2d 2703	. . 3 |- (x = y -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` y)))
8	5, 7	imbi12d 629	. 2 |- (x = y -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. y -> (aleph` A) ~< (aleph` y))))
9		eleq2 1578	. . 3 |- (x = suc y -> (A e. x <-> A e. suc y))
10		fveq2 3835	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
11	10	breq2d 2703	. . 3 |- (x = suc y -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` suc y)))
12	9, 11	imbi12d 629	. 2 |- (x = suc y -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. suc y -> (aleph` A) ~< (aleph` suc y))))
13		eleq2 1578	. . 3 |- (x = B -> (A e. x <-> A e. B))
14		fveq2 3835	. . . 4 |- (x = B -> (aleph` x) = (aleph` B))
15	14	breq2d 2703	. . 3 |- (x = B -> ((aleph` A) ~< (aleph` x) <-> (aleph` A) ~< (aleph` B)))
16	13, 15	imbi12d 629	. 2 |- (x = B -> ((A e. x -> (aleph` A) ~< (aleph` x)) <-> (A e. B -> (aleph` A) ~< (aleph` B))))
17		noel 2336	. . 3 |- -. A e. (/)
18	17	pm2.21i 77	. 2 |- (A e. (/) -> (aleph` A) ~< (aleph` (/)))
19		sdomtr 4619	. . . . . . . . 9 |- (((aleph` A) ~< (aleph` y) /\ (aleph` y) ~< (aleph` suc y)) -> (aleph` A) ~< (aleph` suc y))
20		alephordlem1 5022	. . . . . . . . 9 |- (y e. On -> (aleph` y) ~< (aleph` suc y))
21	19, 20	sylan2 453	. . . . . . . 8 |- (((aleph` A) ~< (aleph` y) /\ y e. On) -> (aleph` A) ~< (aleph` suc y))
22	21	expcom 372	. . . . . . 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 3835	. . . . . . . . 9 |- (A = y -> (aleph` A) = (aleph` y))
26	25	breq1d 2702	. . . . . . . 8 |- (A = y -> ((aleph` A) ~< (aleph` suc y) <-> (aleph` y) ~< (aleph` suc y)))
27	26, 20	syl5bir 208	. . . . . . 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 1859	. . . . 5 |- y e. V
32	31	elsuc2 3043	. . . 4 |- (A e. suc y <-> (A e. y \/ A = y))
33	30, 32	syl5ib 204	. . 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 1859	. . . . . . . . 9 |- x e. V
36		alephlim 5014	. . . . . . . . 9 |- ((x e. V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
37	35, 36	mpan 699	. . . . . . . 8 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
38	37	sseq2d 2141	. . . . . . 7 |- (Lim x -> ((aleph` A) (_ (aleph` x) <-> (aleph` A) (_ U_y e. x (aleph` y)))
39		fveq2 3835	. . . . . . . 8 |- (y = A -> (aleph` y) = (aleph` A))
40	39	ssiun2s 2662	. . . . . . 7 |- (A e. x -> (aleph` A) (_ U_y e. x (aleph` y))
41	38, 40	syl5bir 208	. . . . . 6 |- (Lim x -> (A e. x -> (aleph` A) (_ (aleph` x)))
42		alephon 5015	. . . . . . 7 |- (aleph` A) e. On
43		ssdomg 4549	. . . . . . 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 3203	. . . . . . . . . 10 |- (Lim x -> (A e. x <-> suc A e. x))
47		alephordlem2 5023	. . . . . . . . . . 11 |- ((x e. V /\ Lim x) -> (suc A e. x -> (aleph` suc A) ~<_ (aleph` x)))
48	35, 47	mpan 699	. . . . . . . . . 10 |- (Lim x -> (suc A e. x -> (aleph` suc A) ~<_ (aleph` x)))
49	46, 48	sylbid 201	. . . . . . . . 9 |- (Lim x -> (A e. x -> (aleph` suc A) ~<_ (aleph` x)))
50	49	imp 348	. . . . . . . 8 |- ((Lim x /\ A e. x) -> (aleph` suc A) ~<_ (aleph` x))
51		domnsym 4608	. . . . . . . 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 3843	. . . . . . . . . . 11 |- (aleph` x) e. V
54	53	ensym 4553	. . . . . . . . . 10 |- ((aleph` A) ~~ (aleph` x) -> (aleph` x) ~~ (aleph` A))
55		ensdomtr 4616	. . . . . . . . . . 11 |- (((aleph` x) ~~ (aleph` A) /\ (aleph` A) ~< (aleph` suc A)) -> (aleph` x) ~< (aleph` suc A))
56	55	ex 371	. . . . . . . . . 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 5022	. . . . . . . . 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 3000	. . . . . . . . 9 |- ((x e. On /\ A e. x) -> A e. On)
61		limelon 3036	. . . . . . . . . 10 |- ((x e. V /\ Lim x) -> x e. On)
62	35, 61	mpan 699	. . . . . . . . 9 |- (Lim x -> x e. On)
63	60, 62	sylan 450	. . . . . . . 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 107	. . . . . 6 |- ((Lim x /\ A e. x) -> -. (aleph` A) ~~ (aleph` x))
66	65	ex 371	. . . . 5 |- (Lim x -> (A e. x -> -. (aleph` A) ~~ (aleph` x)))
67	45, 66	jcad 603	. . . 4 |- (Lim x -> (A e. x -> ((aleph` A) ~<_ (aleph` x) /\ -. (aleph` A) ~~ (aleph` x))))
68		brsdom 4522	. . . 4 |- ((aleph` A) ~< (aleph` x) <-> ((aleph` A) ~<_ (aleph` x) /\ -. (aleph` A) ~~ (aleph` x)))
69	67, 68	syl6ibr 211	. . 3 |- (Lim x -> (A e. x -> (aleph` A) ~< (aleph` x)))
70	69	a1d 12	. 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 3212	1 |- (B e. On -> (A e. B -> (aleph` A) ~< (aleph` B)))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994  A.wral 1691  Vcvv 1857   (_ wss 2099  (/)c0 2332  U_ciun 2633   class class class wbr 2692  Oncon0 2975  Lim wlim 2976  suc csuc 2977  ` cfv 3263   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507  alephcale 4960

This theorem is referenced by:  alephord 5025  alephval2 5052

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-rdg 4233  df-er 4401  df-en 4509  df-dom 4510  df-sdom 4511  df-aleph 4963

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