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Theorem alephle 7648
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 7669, 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
StepHypRef Expression
1 id 21 . . 3  |-  ( x  =  y  ->  x  =  y )
2 fveq2 5423 . . 3  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
31, 2sseq12d 3149 . 2  |-  ( x  =  y  ->  (
x  C_  ( aleph `  x )  <->  y  C_  ( aleph `  y )
) )
4 id 21 . . 3  |-  ( x  =  A  ->  x  =  A )
5 fveq2 5423 . . 3  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
64, 5sseq12d 3149 . 2  |-  ( x  =  A  ->  (
x  C_  ( aleph `  x )  <->  A  C_  ( aleph `  A ) ) )
7 alephord2i 7637 . . . . . 6  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( aleph `  y )  e.  ( aleph `  x )
) )
87imp 420 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( aleph `  y )  e.  ( aleph `  x )
)
9 onelon 4354 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
10 alephon 7629 . . . . . 6  |-  ( aleph `  x )  e.  On
11 ontr2 4376 . . . . . 6  |-  ( ( y  e.  On  /\  ( aleph `  x )  e.  On )  ->  (
( y  C_  ( aleph `  y )  /\  ( aleph `  y )  e.  ( aleph `  x )
)  ->  y  e.  ( aleph `  x )
) )
129, 10, 11sylancl 646 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( ( y  C_  ( aleph `  y )  /\  ( aleph `  y )  e.  ( aleph `  x )
)  ->  y  e.  ( aleph `  x )
) )
138, 12mpan2d 658 . . . 4  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( y  C_  ( aleph `  y )  -> 
y  e.  ( aleph `  x ) ) )
1413ralimdva 2592 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  y  C_  ( aleph `  y
)  ->  A. y  e.  x  y  e.  ( aleph `  x )
) )
1510onirri 4436 . . . . 5  |-  -.  ( aleph `  x )  e.  ( aleph `  x )
16 eleq1 2316 . . . . . 6  |-  ( y  =  ( aleph `  x
)  ->  ( y  e.  ( aleph `  x )  <->  (
aleph `  x )  e.  ( aleph `  x )
) )
1716rcla4cv 2832 . . . . 5  |-  ( A. y  e.  x  y  e.  ( aleph `  x )  ->  ( ( aleph `  x
)  e.  x  -> 
( aleph `  x )  e.  ( aleph `  x )
) )
1815, 17mtoi 171 . . . 4  |-  ( A. y  e.  x  y  e.  ( aleph `  x )  ->  -.  ( aleph `  x
)  e.  x )
19 ontri1 4363 . . . . 5  |-  ( ( x  e.  On  /\  ( aleph `  x )  e.  On )  ->  (
x  C_  ( aleph `  x )  <->  -.  ( aleph `  x )  e.  x ) )
2010, 19mpan2 655 . . . 4  |-  ( x  e.  On  ->  (
x  C_  ( aleph `  x )  <->  -.  ( aleph `  x )  e.  x ) )
2118, 20syl5ibr 214 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  y  e.  ( aleph `  x )  ->  x  C_  ( aleph `  x )
) )
2214, 21syld 42 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  y  C_  ( aleph `  y
)  ->  x  C_  ( aleph `  x ) ) )
233, 6, 22tfis3 4585 1  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516    C_ wss 3094   Oncon0 4329   ` cfv 4638   alephcale 7502
This theorem is referenced by:  cardaleph  7649  alephfp  7668  winafp  8252
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-har 7205  df-card 7505  df-aleph 7506
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