Theorem alephle 6032

Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 6049, we will that equality can sometimes hold.)

Assertion

Ref	Expression
alephle	|- (A e. On -> A C_ (aleph` A))

Proof of Theorem alephle

Step	Hyp	Ref	Expression
1		id 73	. . 3 |- (x = y -> x = y)
2		fveq2 4681	. . 3 |- (x = y -> (aleph` x) = (aleph` y))
3	1, 2	sseq12d 2646	. 2 |- (x = y -> (x C_ (aleph` x) <-> y C_ (aleph` y)))
4		id 73	. . 3 |- (x = A -> x = A)
5		fveq2 4681	. . 3 |- (x = A -> (aleph` x) = (aleph` A))
6	4, 5	sseq12d 2646	. 2 |- (x = A -> (x C_ (aleph` x) <-> A C_ (aleph` A)))
7		alephord2i 6025	. . . . . 6 |- (x e. On -> (y e. x -> (aleph` y) e. (aleph` x)))
8	7	imp 377	. . . . 5 |- ((x e. On /\ y e. x) -> (aleph` y) e. (aleph` x))
9		ontr2 3711	. . . . . 6 |- ((y e. On /\ (aleph` x) e. On) -> ((y C_ (aleph` y) /\ (aleph` y) e. (aleph` x)) -> y e. (aleph` x)))
10		onelon 3683	. . . . . 6 |- ((x e. On /\ y e. x) -> y e. On)
11		alephon 5876	. . . . . 6 |- (aleph` x) e. On
12	9, 10, 11	sylancl 525	. . . . 5 |- ((x e. On /\ y e. x) -> ((y C_ (aleph` y) /\ (aleph` y) e. (aleph` x)) -> y e. (aleph` x)))
13	8, 12	mpan2d 766	. . . 4 |- ((x e. On /\ y e. x) -> (y C_ (aleph` y) -> y e. (aleph` x)))
14	13	ralimdvaa 2171	. . 3 |- (x e. On -> (A.y e. x y C_ (aleph` y) -> A.y e. x y e. (aleph` x)))
15		ontri1 3695	. . . . 5 |- ((x e. On /\ (aleph` x) e. On) -> (x C_ (aleph` x) <-> -. (aleph` x) e. x))
16	11, 15	mpan2 760	. . . 4 |- (x e. On -> (x C_ (aleph` x) <-> -. (aleph` x) e. x))
17		elirr 5701	. . . . 5 |- -. (aleph` x) e. (aleph` x)
18		eleq1 1957	. . . . . 6 |- (y = (aleph` x) -> (y e. (aleph` x) <-> (aleph` x) e. (aleph` x)))
19	18	rcla4cv 2377	. . . . 5 |- (A.y e. x y e. (aleph` x) -> ((aleph` x) e. x -> (aleph` x) e. (aleph` x)))
20	17, 19	mtoi 122	. . . 4 |- (A.y e. x y e. (aleph` x) -> -. (aleph` x) e. x)
21	16, 20	syl5bir 227	. . 3 |- (x e. On -> (A.y e. x y e. (aleph` x) -> x C_ (aleph` x)))
22	14, 21	syld 30	. 2 |- (x e. On -> (A.y e. x y C_ (aleph` y) -> x C_ (aleph` x)))
23	3, 6, 22	tfis3 3941	1 |- (A e. On -> A C_ (aleph` A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  Oncon0 3657  ` cfv 3998  alephcale 5860

This theorem is referenced by:  cardaleph 6033  alephfp 6048

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-reg 5695  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-fin 5430  df-card 5862  df-aleph 5863

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