Theorem alephordi 8897

Description: Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.)

Assertion

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

Proof of Theorem alephordi

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

Step	Hyp	Ref	Expression
1		eleq2 2690	. . 3 |- ( x = (/) -> ( A e. x <-> A e. (/) ) )
2		fveq2 6191	. . . 4 |- ( x = (/) -> ( aleph ` x ) = ( aleph ` (/) ) )
3	2	breq2d 4665	. . 3 |- ( x = (/) -> ( ( aleph ` A ) ~< ( aleph ` x ) <-> ( aleph ` A ) ~< ( aleph ` (/) ) ) )
4	1, 3	imbi12d 334	. 2 |- ( x = (/) -> ( ( A e. x -> ( aleph ` A ) ~< ( aleph ` x ) ) <-> ( A e. (/) -> ( aleph ` A ) ~< ( aleph ` (/) ) ) ) )
5		eleq2 2690	. . 3 |- ( x = y -> ( A e. x <-> A e. y ) )
6		fveq2 6191	. . . 4 |- ( x = y -> ( aleph ` x ) = ( aleph ` y ) )
7	6	breq2d 4665	. . 3 |- ( x = y -> ( ( aleph ` A ) ~< ( aleph ` x ) <-> ( aleph ` A ) ~< ( aleph ` y ) ) )
8	5, 7	imbi12d 334	. 2 |- ( x = y -> ( ( A e. x -> ( aleph ` A ) ~< ( aleph ` x ) ) <-> ( A e. y -> ( aleph ` A ) ~< ( aleph ` y ) ) ) )
9		eleq2 2690	. . 3 |- ( x = suc y -> ( A e. x <-> A e. suc y ) )
10		fveq2 6191	. . . 4 |- ( x = suc y -> ( aleph ` x ) = ( aleph ` suc y ) )
11	10	breq2d 4665	. . 3 |- ( x = suc y -> ( ( aleph ` A ) ~< ( aleph ` x ) <-> ( aleph ` A ) ~< ( aleph ` suc y ) ) )
12	9, 11	imbi12d 334	. 2 |- ( x = suc y -> ( ( A e. x -> ( aleph ` A ) ~< ( aleph ` x ) ) <-> ( A e. suc y -> ( aleph ` A ) ~< ( aleph ` suc y ) ) ) )
13		eleq2 2690	. . 3 |- ( x = B -> ( A e. x <-> A e. B ) )
14		fveq2 6191	. . . 4 |- ( x = B -> ( aleph ` x ) = ( aleph ` B ) )
15	14	breq2d 4665	. . 3 |- ( x = B -> ( ( aleph ` A ) ~< ( aleph ` x ) <-> ( aleph ` A ) ~< ( aleph ` B ) ) )
16	13, 15	imbi12d 334	. 2 |- ( x = B -> ( ( A e. x -> ( aleph ` A ) ~< ( aleph ` x ) ) <-> ( A e. B -> ( aleph ` A ) ~< ( aleph ` B ) ) ) )
17		noel 3919	. . 3 |- -. A e. (/)
18	17	pm2.21i 116	. 2 |- ( A e. (/) -> ( aleph ` A ) ~< ( aleph ` (/) ) )
19		vex 3203	. . . . 5 |- y e. _V
20	19	elsuc2 5795	. . . 4 |- ( A e. suc y <-> ( A e. y \/ A = y ) )
21		alephordilem1 8896	. . . . . . . . 9 |- ( y e. On -> ( aleph ` y ) ~< ( aleph ` suc y ) )
22		sdomtr 8098	. . . . . . . . 9 |- ( ( ( aleph ` A ) ~< ( aleph ` y ) /\ ( aleph ` y ) ~< ( aleph ` suc y ) ) -> ( aleph ` A ) ~< ( aleph ` suc y ) )
23	21, 22	sylan2 491	. . . . . . . 8 |- ( ( ( aleph ` A ) ~< ( aleph ` y ) /\ y e. On ) -> ( aleph ` A ) ~< ( aleph ` suc y ) )
24	23	expcom 451	. . . . . . 7 |- ( y e. On -> ( ( aleph ` A ) ~< ( aleph ` y ) -> ( aleph ` A ) ~< ( aleph ` suc y ) ) )
25	24	imim2d 57	. . . . . 6 |- ( y e. On -> ( ( A e. y -> ( aleph ` A ) ~< ( aleph ` y ) ) -> ( A e. y -> ( aleph ` A ) ~< ( aleph ` suc y ) ) ) )
26	25	com23 86	. . . . 5 |- ( y e. On -> ( A e. y -> ( ( A e. y -> ( aleph ` A ) ~< ( aleph ` y ) ) -> ( aleph ` A ) ~< ( aleph ` suc y ) ) ) )
27		fveq2 6191	. . . . . . . . 9 |- ( A = y -> ( aleph ` A ) = ( aleph ` y ) )
28	27	breq1d 4663	. . . . . . . 8 |- ( A = y -> ( ( aleph ` A ) ~< ( aleph ` suc y ) <-> ( aleph ` y ) ~< ( aleph ` suc y ) ) )
29	21, 28	syl5ibr 236	. . . . . . 7 |- ( A = y -> ( y e. On -> ( aleph ` A ) ~< ( aleph ` suc y ) ) )
30	29	a1d 25	. . . . . 6 |- ( A = y -> ( ( A e. y -> ( aleph ` A ) ~< ( aleph ` y ) ) -> ( y e. On -> ( aleph ` A ) ~< ( aleph ` suc y ) ) ) )
31	30	com3r 87	. . . . 5 |- ( y e. On -> ( A = y -> ( ( A e. y -> ( aleph ` A ) ~< ( aleph ` y ) ) -> ( aleph ` A ) ~< ( aleph ` suc y ) ) ) )
32	26, 31	jaod 395	. . . 4 |- ( y e. On -> ( ( A e. y \/ A = y ) -> ( ( A e. y -> ( aleph ` A ) ~< ( aleph ` y ) ) -> ( aleph ` A ) ~< ( aleph ` suc y ) ) ) )
33	20, 32	syl5bi 232	. . 3 |- ( y e. On -> ( A e. suc y -> ( ( A e. y -> ( aleph ` A ) ~< ( aleph ` y ) ) -> ( aleph ` A ) ~< ( aleph ` suc y ) ) ) )
34	33	com23 86	. 2 |- ( y e. On -> ( ( A e. y -> ( aleph ` A ) ~< ( aleph ` y ) ) -> ( A e. suc y -> ( aleph ` A ) ~< ( aleph ` suc y ) ) ) )
35		fvexd 6203	. . . . . 6 |- ( Lim x -> ( aleph ` x ) e. _V )
36		fveq2 6191	. . . . . . . 8 |- ( w = A -> ( aleph ` w ) = ( aleph ` A ) )
37	36	ssiun2s 4564	. . . . . . 7 |- ( A e. x -> ( aleph ` A ) C_ U_ w e. x ( aleph ` w ) )
38		vex 3203	. . . . . . . . 9 |- x e. _V
39		alephlim 8890	. . . . . . . . 9 |- ( ( x e. _V /\ Lim x ) -> ( aleph ` x ) = U_ w e. x ( aleph ` w ) )
40	38, 39	mpan 706	. . . . . . . 8 |- ( Lim x -> ( aleph ` x ) = U_ w e. x ( aleph ` w ) )
41	40	sseq2d 3633	. . . . . . 7 |- ( Lim x -> ( ( aleph ` A ) C_ ( aleph ` x ) <-> ( aleph ` A ) C_ U_ w e. x ( aleph ` w ) ) )
42	37, 41	syl5ibr 236	. . . . . 6 |- ( Lim x -> ( A e. x -> ( aleph ` A ) C_ ( aleph ` x ) ) )
43		ssdomg 8001	. . . . . 6 |- ( ( aleph ` x ) e. _V -> ( ( aleph ` A ) C_ ( aleph ` x ) -> ( aleph ` A ) ~<_ ( aleph ` x ) ) )
44	35, 42, 43	sylsyld 61	. . . . 5 |- ( Lim x -> ( A e. x -> ( aleph ` A ) ~<_ ( aleph ` x ) ) )
45		limsuc 7049	. . . . . . . . . 10 |- ( Lim x -> ( A e. x <-> suc A e. x ) )
46		fveq2 6191	. . . . . . . . . . . . 13 |- ( w = suc A -> ( aleph ` w ) = ( aleph ` suc A ) )
47	46	ssiun2s 4564	. . . . . . . . . . . 12 |- ( suc A e. x -> ( aleph ` suc A ) C_ U_ w e. x ( aleph ` w ) )
48	40	sseq2d 3633	. . . . . . . . . . . 12 |- ( Lim x -> ( ( aleph ` suc A ) C_ ( aleph ` x ) <-> ( aleph ` suc A ) C_ U_ w e. x ( aleph ` w ) ) )
49	47, 48	syl5ibr 236	. . . . . . . . . . 11 |- ( Lim x -> ( suc A e. x -> ( aleph ` suc A ) C_ ( aleph ` x ) ) )
50		ssdomg 8001	. . . . . . . . . . 11 |- ( ( aleph ` x ) e. _V -> ( ( aleph ` suc A ) C_ ( aleph ` x ) -> ( aleph ` suc A ) ~<_ ( aleph ` x ) ) )
51	35, 49, 50	sylsyld 61	. . . . . . . . . 10 |- ( Lim x -> ( suc A e. x -> ( aleph ` suc A ) ~<_ ( aleph ` x ) ) )
52	45, 51	sylbid 230	. . . . . . . . 9 |- ( Lim x -> ( A e. x -> ( aleph ` suc A ) ~<_ ( aleph ` x ) ) )
53	52	imp 445	. . . . . . . 8 |- ( ( Lim x /\ A e. x ) -> ( aleph ` suc A ) ~<_ ( aleph ` x ) )
54		domnsym 8086	. . . . . . . 8 |- ( ( aleph ` suc A ) ~<_ ( aleph ` x ) -> -. ( aleph ` x ) ~< ( aleph ` suc A ) )
55	53, 54	syl 17	. . . . . . 7 |- ( ( Lim x /\ A e. x ) -> -. ( aleph ` x ) ~< ( aleph ` suc A ) )
56		limelon 5788	. . . . . . . . . 10 |- ( ( x e. _V /\ Lim x ) -> x e. On )
57	38, 56	mpan 706	. . . . . . . . 9 |- ( Lim x -> x e. On )
58		onelon 5748	. . . . . . . . 9 |- ( ( x e. On /\ A e. x ) -> A e. On )
59	57, 58	sylan 488	. . . . . . . 8 |- ( ( Lim x /\ A e. x ) -> A e. On )
60		ensym 8005	. . . . . . . . 9 |- ( ( aleph ` A ) ~~ ( aleph ` x ) -> ( aleph ` x ) ~~ ( aleph ` A ) )
61		alephordilem1 8896	. . . . . . . . 9 |- ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) )
62		ensdomtr 8096	. . . . . . . . . 10 |- ( ( ( aleph ` x ) ~~ ( aleph ` A ) /\ ( aleph ` A ) ~< ( aleph ` suc A ) ) -> ( aleph ` x ) ~< ( aleph ` suc A ) )
63	62	ex 450	. . . . . . . . 9 |- ( ( aleph ` x ) ~~ ( aleph ` A ) -> ( ( aleph ` A ) ~< ( aleph ` suc A ) -> ( aleph ` x ) ~< ( aleph ` suc A ) ) )
64	60, 61, 63	syl2im 40	. . . . . . . 8 |- ( ( aleph ` A ) ~~ ( aleph ` x ) -> ( A e. On -> ( aleph ` x ) ~< ( aleph ` suc A ) ) )
65	59, 64	syl5com 31	. . . . . . 7 |- ( ( Lim x /\ A e. x ) -> ( ( aleph ` A ) ~~ ( aleph ` x ) -> ( aleph ` x ) ~< ( aleph ` suc A ) ) )
66	55, 65	mtod 189	. . . . . 6 |- ( ( Lim x /\ A e. x ) -> -. ( aleph ` A ) ~~ ( aleph ` x ) )
67	66	ex 450	. . . . 5 |- ( Lim x -> ( A e. x -> -. ( aleph ` A ) ~~ ( aleph ` x ) ) )
68	44, 67	jcad 555	. . . 4 |- ( Lim x -> ( A e. x -> ( ( aleph ` A ) ~<_ ( aleph ` x ) /\ -. ( aleph ` A ) ~~ ( aleph ` x ) ) ) )
69		brsdom 7978	. . . 4 |- ( ( aleph ` A ) ~< ( aleph ` x ) <-> ( ( aleph ` A ) ~<_ ( aleph ` x ) /\ -. ( aleph ` A ) ~~ ( aleph ` x ) ) )
70	68, 69	syl6ibr 242	. . 3 |- ( Lim x -> ( A e. x -> ( aleph ` A ) ~< ( aleph ` x ) ) )
71	70	a1d 25	. 2 |- ( Lim x -> ( A. y e. x ( A e. y -> ( aleph ` A ) ~< ( aleph ` y ) ) -> ( A e. x -> ( aleph ` A ) ~< ( aleph ` x ) ) ) )
72	4, 8, 12, 16, 18, 34, 71	tfinds 7059	1 |- ( B e. On -> ( A e. B -> ( aleph ` A ) ~< ( aleph ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U_ciun 4520   class class class wbr 4653   Oncon0 5723   Lim wlim 5724   suc csuc 5725   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   alephcale 8762

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

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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by:  alephord  8898  alephval2  9394