Theorem alephle 4856

Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 4873, we will that equality can sometimes hold.)

Assertion

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

Proof of Theorem alephle

Step	Hyp	Ref	Expression
1		id 59	. . 3 |- (x = y -> x = y)
2		fveq2 3709	. . 3 |- (x = y -> (aleph` x) = (aleph` y))
3	1, 2	sseq12d 2080	. 2 |- (x = y -> (x (_ (aleph` x) <-> y (_ (aleph` y)))
4		id 59	. . 3 |- (x = A -> x = A)
5		fveq2 3709	. . 3 |- (x = A -> (aleph` x) = (aleph` A))
6	4, 5	sseq12d 2080	. 2 |- (x = A -> (x (_ (aleph` x) <-> A (_ (aleph` A)))
7		alephord2i 4849	. . . . . 6 |- (x e. On -> (y e. x -> (aleph` y) e. (aleph` x)))
8	7	imp 350	. . . . 5 |- ((x e. On /\ y e. x) -> (aleph` y) e. (aleph` x))
9		onelon 2962	. . . . . 6 |- ((x e. On /\ y e. x) -> y e. On)
10		alephon 4837	. . . . . . 7 |- (aleph` x) e. On
11		ontr2 2994	. . . . . . 7 |- ((y e. On /\ (aleph` x) e. On) -> ((y (_ (aleph` y) /\ (aleph` y) e. (aleph` x)) -> y e. (aleph` x)))
12	10, 11	mpan2 694	. . . . . 6 |- (y e. On -> ((y (_ (aleph` y) /\ (aleph` y) e. (aleph` x)) -> y e. (aleph` x)))
13	9, 12	syl 10	. . . . 5 |- ((x e. On /\ y e. x) -> ((y (_ (aleph` y) /\ (aleph` y) e. (aleph` x)) -> y e. (aleph` x)))
14	8, 13	mpan2d 700	. . . 4 |- ((x e. On /\ y e. x) -> (y (_ (aleph` y) -> y e. (aleph` x)))
15	14	r19.20dva 1701	. . 3 |- (x e. On -> (A.y e. x y (_ (aleph` y) -> A.y e. x y e. (aleph` x)))
16		ontri1 2971	. . . . 5 |- ((x e. On /\ (aleph` x) e. On) -> (x (_ (aleph` x) <-> -. (aleph` x) e. x))
17	10, 16	mpan2 694	. . . 4 |- (x e. On -> (x (_ (aleph` x) <-> -. (aleph` x) e. x))
18		elirr 4571	. . . . 5 |- -. (aleph` x) e. (aleph` x)
19		eleq1 1526	. . . . . 6 |- (y = (aleph` x) -> (y e. (aleph` x) <-> (aleph` x) e. (aleph` x)))
20	19	rcla4cv 1865	. . . . 5 |- (A.y e. x y e. (aleph` x) -> ((aleph` x) e. x -> (aleph` x) e. (aleph` x)))
21	18, 20	mtoi 107	. . . 4 |- (A.y e. x y e. (aleph` x) -> -. (aleph` x) e. x)
22	17, 21	syl5bir 210	. . 3 |- (x e. On -> (A.y e. x y e. (aleph` x) -> x (_ (aleph` x)))
23	15, 22	syld 27	. 2 |- (x e. On -> (A.y e. x y (_ (aleph` y) -> x (_ (aleph` x)))
24	3, 6, 23	tfis3 3120	1 |- (A e. On -> A (_ (aleph` A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637   (_ wss 2037  Oncon0 2938  ` cfv 3172  alephcale 4786

This theorem is referenced by:  cardaleph 4857  alephfp 4872

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789

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