Theorem alephle 4895

Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 4912, 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 3730	. . 3 |- (x = y -> (aleph` x) = (aleph` y))
3	1, 2	sseq12d 2093	. 2 |- (x = y -> (x (_ (aleph` x) <-> y (_ (aleph` y)))
4		id 59	. . 3 |- (x = A -> x = A)
5		fveq2 3730	. . 3 |- (x = A -> (aleph` x) = (aleph` A))
6	4, 5	sseq12d 2093	. 2 |- (x = A -> (x (_ (aleph` x) <-> A (_ (aleph` A)))
7		alephord2i 4888	. . . . . 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 2978	. . . . . 6 |- ((x e. On /\ y e. x) -> y e. On)
10		alephon 4876	. . . . . . 7 |- (aleph` x) e. On
11		ontr2 3010	. . . . . . 7 |- ((y e. On /\ (aleph` x) e. On) -> ((y (_ (aleph` y) /\ (aleph` y) e. (aleph` x)) -> y e. (aleph` x)))
12	10, 11	mpan2 698	. . . . . 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 704	. . . 4 |- ((x e. On /\ y e. x) -> (y (_ (aleph` y) -> y e. (aleph` x)))
15	14	r19.20dva 1712	. . 3 |- (x e. On -> (A.y e. x y (_ (aleph` y) -> A.y e. x y e. (aleph` x)))
16		ontri1 2987	. . . . 5 |- ((x e. On /\ (aleph` x) e. On) -> (x (_ (aleph` x) <-> -. (aleph` x) e. x))
17	10, 16	mpan2 698	. . . 4 |- (x e. On -> (x (_ (aleph` x) <-> -. (aleph` x) e. x))
18		elirr 4608	. . . . 5 |- -. (aleph` x) e. (aleph` x)
19		eleq1 1537	. . . . . 6 |- (y = (aleph` x) -> (y e. (aleph` x) <-> (aleph` x) e. (aleph` x)))
20	19	rcla4cv 1877	. . . . 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 3136	1 |- (A e. On -> A (_ (aleph` A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648   (_ wss 2050  Oncon0 2954  ` cfv 3188  alephcale 4824

This theorem is referenced by:  cardaleph 4896  alephfp 4911

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376  df-fin 4377  df-card 4826  df-aleph 4827

Copyright terms: Public domain