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Theorem alephle 7669
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 7690, 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 5444 . . 3  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
31, 2sseq12d 3168 . 2  |-  ( x  =  y  ->  (
x  C_  ( aleph `  x )  <->  y  C_  ( aleph `  y )
) )
4 id 21 . . 3  |-  ( x  =  A  ->  x  =  A )
5 fveq2 5444 . . 3  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
64, 5sseq12d 3168 . 2  |-  ( x  =  A  ->  (
x  C_  ( aleph `  x )  <->  A  C_  ( aleph `  A ) ) )
7 alephord2i 7658 . . . . . 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 4375 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
10 alephon 7650 . . . . . 6  |-  ( aleph `  x )  e.  On
11 ontr2 4397 . . . . . 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 2594 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  y  C_  ( aleph `  y
)  ->  A. y  e.  x  y  e.  ( aleph `  x )
) )
1510onirri 4457 . . . . 5  |-  -.  ( aleph `  x )  e.  ( aleph `  x )
16 eleq1 2316 . . . . . 6  |-  ( y  =  ( aleph `  x
)  ->  ( y  e.  ( aleph `  x )  <->  (
aleph `  x )  e.  ( aleph `  x )
) )
1716rcla4cv 2849 . . . . 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 4384 . . . . 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 4606 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 3113   Oncon0 4350   ` cfv 4659   alephcale 7523
This theorem is referenced by:  cardaleph  7670  alephfp  7689  winafp  8273
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 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296
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 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-oi 7179  df-har 7226  df-card 7526  df-aleph 7527
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