Theorem alephval2 5052

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 2183	. . 3 |- {x e. On | A.y e. A (aleph` y) ~< x} (_ On
2		oneqmini 3024	. . 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 5024	. . . . . 6 |- (A e. On -> (y e. A -> (aleph` y) ~< (aleph` A)))
5	4	r19.21aiv 1759	. . . . 5 |- (A e. On -> A.y e. A (aleph` y) ~< (aleph` A))
6		alephon 5015	. . . . 5 |- (aleph` A) e. On
7	5, 6	jctil 290	. . . 4 |- (A e. On -> ((aleph` A) e. On /\ A.y e. A (aleph` y) ~< (aleph` A)))
8		breq2 2696	. . . . . 6 |- (x = (aleph` A) -> ((aleph` y) ~< x <-> (aleph` y) ~< (aleph` A)))
9	8	ralbidv 1709	. . . . 5 |- (x = (aleph` A) -> (A.y e. A (aleph` y) ~< x <-> A.y e. A (aleph` y) ~< (aleph` A)))
10	9	elrab 1951	. . . 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 198	. . 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 4772	. . . . . . 7 |- om e. V
14		visset 1859	. . . . . . 7 |- z e. V
15		entri3 4990	. . . . . . 7 |- ((om e. V /\ z e. V) -> (om ~<_ z \/ z ~<_ om))
16	13, 14, 15	mp2an 701	. . . . . 6 |- (om ~<_ z \/ z ~<_ om)
17		alephord 5025	. . . . . . . . . . . . . . 15 |- ((x e. On /\ A e. On) -> (x e. A <-> (aleph` x) ~< (aleph` A)))
18	17	ancoms 438	. . . . . . . . . . . . . 14 |- ((A e. On /\ x e. On) -> (x e. A <-> (aleph` x) ~< (aleph` A)))
19		breq1 2695	. . . . . . . . . . . . . . 15 |- ((card` z) = (aleph` x) -> ((card` z) ~< (aleph` A) <-> (aleph` x) ~< (aleph` A)))
20		cardid 4975	. . . . . . . . . . . . . . . 16 |- (card` z) ~~ z
21		sdomen1 4626	. . . . . . . . . . . . . . . 16 |- ((z e. V /\ (card` z) ~~ z) -> ((card` z) ~< (aleph` A) <-> z ~< (aleph` A)))
22	14, 20, 21	mp2an 701	. . . . . . . . . . . . . . 15 |- ((card` z) ~< (aleph` A) <-> z ~< (aleph` A))
23	19, 22	syl5rbbr 538	. . . . . . . . . . . . . 14 |- ((card` z) = (aleph` x) -> ((aleph` x) ~< (aleph` A) <-> z ~< (aleph` A)))
24	18, 23	sylan9bb 543	. . . . . . . . . . . . 13 |- (((A e. On /\ x e. On) /\ (card` z) = (aleph` x)) -> (x e. A <-> z ~< (aleph` A)))
25		fveq2 3835	. . . . . . . . . . . . . . . . . 18 |- (y = x -> (aleph` y) = (aleph` x))
26	25	breq1d 2702	. . . . . . . . . . . . . . . . 17 |- (y = x -> ((aleph` y) ~< z <-> (aleph` x) ~< z))
27	26	rcla4v 1919	. . . . . . . . . . . . . . . 16 |- (x e. A -> (A.y e. A (aleph` y) ~< z -> (aleph` x) ~< z))
28		sdomirr 4617	. . . . . . . . . . . . . . . . 17 |- -. (aleph` x) ~< (aleph` x)
29		breq2 2696	. . . . . . . . . . . . . . . . . 18 |- ((card` z) = (aleph` x) -> ((aleph` x) ~< (card` z) <-> (aleph` x) ~< (aleph` x)))
30		sdomen2 4627	. . . . . . . . . . . . . . . . . . 19 |- ((z e. V /\ (card` z) ~~ z) -> ((aleph` x) ~< (card` z) <-> (aleph` x) ~< z))
31	14, 20, 30	mp2an 701	. . . . . . . . . . . . . . . . . 18 |- ((aleph` x) ~< (card` z) <-> (aleph` x) ~< z)
32	29, 31	syl5bbr 537	. . . . . . . . . . . . . . . . 17 |- ((card` z) = (aleph` x) -> ((aleph` x) ~< z <-> (aleph` x) ~< (aleph` x)))
33	28, 32	mtbiri 722	. . . . . . . . . . . . . . . 16 |- ((card` z) = (aleph` x) -> -. (aleph` x) ~< z)
34	27, 33	nsyli 120	. . . . . . . . . . . . . . 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 203	. . . . . . . . . . . 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 1792	. . . . . . . . . 10 |- (A e. On -> (E.x e. On (card` z) = (aleph` x) -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z)))
40		cardidm 4999	. . . . . . . . . . 11 |- (card` (card` z)) = (card` z)
41		cardalephex 5036	. . . . . . . . . . 11 |- (om (_ (card` z) -> ((card` (card` z)) = (card` z) <-> E.x e. On (card` z) = (aleph` x)))
42	40, 41	mpbii 191	. . . . . . . . . 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 4985	. . . . . . . . . . 11 |- ((om e. V /\ z e. V) -> ((card` om) (_ (card` z) <-> om ~<_ z))
45	13, 14, 44	mp2an 701	. . . . . . . . . 10 |- ((card` om) (_ (card` z) <-> om ~<_ z)
46		cardom 4972	. . . . . . . . . . 11 |- (card` om) = om
47	46	sseq1i 2137	. . . . . . . . . 10 |- ((card` om) (_ (card` z) <-> om (_ (card` z))
48	45, 47	bitr3i 173	. . . . . . . . 9 |- (om ~<_ z <-> om (_ (card` z))
49	43, 48	syl5ib 204	. . . . . . . 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 2338	. . . . . . . . . . . . 13 |- ((/) e. A -> A =/= (/))
52		r19.2z 2401	. . . . . . . . . . . . . . 15 |- ((A =/= (/) /\ A.y e. A -. (aleph` y) ~< z) -> E.y e. A -. (aleph` y) ~< z)
53	52	ex 371	. . . . . . . . . . . . . 14 |- (A =/= (/) -> (A.y e. A -. (aleph` y) ~< z -> E.y e. A -. (aleph` y) ~< z))
54		domtr 4556	. . . . . . . . . . . . . . . . . . . 20 |- ((z ~<_ om /\ om ~<_ (aleph` y)) -> z ~<_ (aleph` y))
55		alephgeom 5032	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. On <-> om (_ (aleph` y))
56		ssdomg 4549	. . . . . . . . . . . . . . . . . . . . . 22 |- (om e. V -> (om (_ (aleph` y) -> om ~<_ (aleph` y)))
57	13, 56	ax-mp 7	. . . . . . . . . . . . . . . . . . . . 21 |- (om (_ (aleph` y) -> om ~<_ (aleph` y))
58	55, 57	sylbi 197	. . . . . . . . . . . . . . . . . . . 20 |- (y e. On -> om ~<_ (aleph` y))
59	54, 58	sylan2 453	. . . . . . . . . . . . . . . . . . 19 |- ((z ~<_ om /\ y e. On) -> z ~<_ (aleph` y))
60		domnsym 4608	. . . . . . . . . . . . . . . . . . 19 |- (z ~<_ (aleph` y) -> -. (aleph` y) ~< z)
61	59, 60	syl 10	. . . . . . . . . . . . . . . . . 18 |- ((z ~<_ om /\ y e. On) -> -. (aleph` y) ~< z)
62		onelon 3000	. . . . . . . . . . . . . . . . . 18 |- ((A e. On /\ y e. A) -> y e. On)
63	61, 62	sylan2 453	. . . . . . . . . . . . . . . . 17 |- ((z ~<_ om /\ (A e. On /\ y e. A)) -> -. (aleph` y) ~< z)
64	63	exp32 377	. . . . . . . . . . . . . . . 16 |- (z ~<_ om -> (A e. On -> (y e. A -> -. (aleph` y) ~< z)))
65	64	imp 348	. . . . . . . . . . . . . . 15 |- ((z ~<_ om /\ A e. On) -> (y e. A -> -. (aleph` y) ~< z))
66	65	r19.21aiv 1759	. . . . . . . . . . . . . 14 |- ((z ~<_ om /\ A e. On) -> A.y e. A -. (aleph` y) ~< z)
67	53, 66	syl5 21	. . . . . . . . . . . . 13 |- (A =/= (/) -> ((z ~<_ om /\ A e. On) -> E.y e. A -. (aleph` y) ~< z))
68	51, 67	syl 10	. . . . . . . . . . . 12 |- ((/) e. A -> ((z ~<_ om /\ A e. On) -> E.y e. A -. (aleph` y) ~< z))
69		rexnal 1700	. . . . . . . . . . . 12 |- (E.y e. A -. (aleph` y) ~< z <-> -. A.y e. A (aleph` y) ~< z)
70	68, 69	syl6ib 210	. . . . . . . . . . 11 |- ((/) e. A -> ((z ~<_ om /\ A e. On) -> -. A.y e. A (aleph` y) ~< z))
71	70	com12 11	. . . . . . . . . 10 |- ((z ~<_ om /\ A e. On) -> ((/) e. A -> -. A.y e. A (aleph` y) ~< z))
72	71	expimpd 373	. . . . . . . . 9 |- (z ~<_ om -> ((A e. On /\ (/) e. A) -> -. A.y e. A (aleph` y) ~< z))
73	72	a1d 12	. . . . . . . 8 |- (z ~<_ om -> (z ~< (aleph` A) -> ((A e. On /\ (/) e. A) -> -. A.y e. A (aleph` y) ~< z)))
74	73	com3r 35	. . . . . . 7 |- ((A e. On /\ (/) e. A) -> (z ~<_ om -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z)))
75	50, 74	jaod 424	. . . . . 6 |- ((A e. On /\ (/) e. A) -> ((om ~<_ z \/ z ~<_ om) -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z)))
76	16, 75	mpi 44	. . . . 5 |- ((A e. On /\ (/) e. A) -> (z ~< (aleph` A) -> -. A.y e. A (aleph` y) ~< z))
77	6	oneli 3077	. . . . . . 7 |- (z e. (aleph` A) -> z e. On)
78		cardsdomel 5002	. . . . . . . 8 |- (z e. On -> (z ~< (aleph` A) <-> z e. (card` (aleph` A))))
79		alephcard 5017	. . . . . . . . 9 |- (card` (aleph` A)) = (aleph` A)
80	79	eleq2i 1581	. . . . . . . 8 |- (z e. (card` (aleph` A)) <-> z e. (aleph` A))
81	78, 80	syl6rbb 540	. . . . . . 7 |- (z e. On -> (z e. (aleph` A) <-> z ~< (aleph` A)))
82	77, 81	syl 10	. . . . . 6 |- (z e. (aleph` A) -> (z e. (aleph` A) <-> z ~< (aleph` A)))
83	82	ibi 595	. . . . 5 |- (z e. (aleph` A) -> z ~< (aleph` A))
84	76, 83	syl5 21	. . . 4 |- ((A e. On /\ (/) e. A) -> (z e. (aleph` A) -> -. A.y e. A (aleph` y) ~< z))
85		breq2 2696	. . . . . . . 8 |- (x = z -> ((aleph` y) ~< x <-> (aleph` y) ~< z))
86	85	ralbidv 1709	. . . . . . 7 |- (x = z -> (A.y e. A (aleph` y) ~< x <-> A.y e. A (aleph` y) ~< z))
87	86	elrab 1951	. . . . . 6 |- (z e. {x e. On | A.y e. A (aleph` y) ~< x} <-> (z e. On /\ A.y e. A (aleph` y) ~< z))
88	87	pm3.27bi 324	. . . . 5 |- (z e. {x e. On | A.y e. A (aleph` y) ~< x} -> A.y e. A (aleph` y) ~< z)
89	88	con3i 98	. . . 4 |- (-. A.y e. A (aleph` y) ~< z -> -. z e. {x e. On | A.y e. A (aleph` y) ~< x})
90	84, 89	syl6 22	. . 3 |- ((A e. On /\ (/) e. A) -> (z e. (aleph` A) -> -. z e. {x e. On | A.y e. A (aleph` y) ~< x}))
91	90	r19.21aiv 1759	. 2 |- ((A e. On /\ (/) e. A) -> A.z e. (aleph` A) -. z e. {x e. On | A.y e. A (aleph` y) ~< x})
92	3, 12, 91	sylanc 473	1 |- ((A e. On /\ (/) e. A) -> (aleph` A) = |^|{x e. On | A.y e. A (aleph` y) ~< x})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994   =/= wne 1628  A.wral 1691  E.wrex 1692  {crab 1694  Vcvv 1857   (_ wss 2099  (/)c0 2332  |^|cint 2600   class class class wbr 2692  Oncon0 2975  omcom 3218  ` cfv 3263   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507  cardccrd 4959  alephcale 4960

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-reg 4736  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-pss 2107  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-fin 4512  df-card 4962  df-aleph 4963

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