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Theorem alephle 8465
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 8486, we will that equality can sometimes hold.) (Contributed by NM, 9-Nov-2003.) (Proof shortened by Mario Carneiro, 22-Feb-2013.)
Assertion
Ref Expression
alephle  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)

Proof of Theorem alephle
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3  |-  ( x  =  y  ->  x  =  y )
2 fveq2 5864 . . 3  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
31, 2sseq12d 3533 . 2  |-  ( x  =  y  ->  (
x  C_  ( aleph `  x )  <->  y  C_  ( aleph `  y )
) )
4 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
5 fveq2 5864 . . 3  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
64, 5sseq12d 3533 . 2  |-  ( x  =  A  ->  (
x  C_  ( aleph `  x )  <->  A  C_  ( aleph `  A ) ) )
7 alephord2i 8454 . . . . . 6  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( aleph `  y )  e.  ( aleph `  x )
) )
87imp 429 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( aleph `  y )  e.  ( aleph `  x )
)
9 onelon 4903 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
10 alephon 8446 . . . . . 6  |-  ( aleph `  x )  e.  On
11 ontr2 4925 . . . . . 6  |-  ( ( y  e.  On  /\  ( aleph `  x )  e.  On )  ->  (
( y  C_  ( aleph `  y )  /\  ( aleph `  y )  e.  ( aleph `  x )
)  ->  y  e.  ( aleph `  x )
) )
129, 10, 11sylancl 662 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( ( y  C_  ( aleph `  y )  /\  ( aleph `  y )  e.  ( aleph `  x )
)  ->  y  e.  ( aleph `  x )
) )
138, 12mpan2d 674 . . . 4  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( y  C_  ( aleph `  y )  -> 
y  e.  ( aleph `  x ) ) )
1413ralimdva 2872 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  y  C_  ( aleph `  y
)  ->  A. y  e.  x  y  e.  ( aleph `  x )
) )
1510onirri 4984 . . . . 5  |-  -.  ( aleph `  x )  e.  ( aleph `  x )
16 eleq1 2539 . . . . . 6  |-  ( y  =  ( aleph `  x
)  ->  ( y  e.  ( aleph `  x )  <->  (
aleph `  x )  e.  ( aleph `  x )
) )
1716rspccv 3211 . . . . 5  |-  ( A. y  e.  x  y  e.  ( aleph `  x )  ->  ( ( aleph `  x
)  e.  x  -> 
( aleph `  x )  e.  ( aleph `  x )
) )
1815, 17mtoi 178 . . . 4  |-  ( A. y  e.  x  y  e.  ( aleph `  x )  ->  -.  ( aleph `  x
)  e.  x )
19 ontri1 4912 . . . . 5  |-  ( ( x  e.  On  /\  ( aleph `  x )  e.  On )  ->  (
x  C_  ( aleph `  x )  <->  -.  ( aleph `  x )  e.  x ) )
2010, 19mpan2 671 . . . 4  |-  ( x  e.  On  ->  (
x  C_  ( aleph `  x )  <->  -.  ( aleph `  x )  e.  x ) )
2118, 20syl5ibr 221 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  y  e.  ( aleph `  x )  ->  x  C_  ( aleph `  x )
) )
2214, 21syld 44 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  y  C_  ( aleph `  y
)  ->  x  C_  ( aleph `  x ) ) )
233, 6, 22tfis3 6670 1  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   Oncon0 4878   ` cfv 5586   alephcale 8313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-oi 7931  df-har 7980  df-card 8316  df-aleph 8317
This theorem is referenced by:  cardaleph  8466  alephfp  8485  winafp  9071
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