MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephle Structured version   Unicode version

Theorem alephle 8501
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 8522, 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 5849 . . 3  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
31, 2sseq12d 3471 . 2  |-  ( x  =  y  ->  (
x  C_  ( aleph `  x )  <->  y  C_  ( aleph `  y )
) )
4 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
5 fveq2 5849 . . 3  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
64, 5sseq12d 3471 . 2  |-  ( x  =  A  ->  (
x  C_  ( aleph `  x )  <->  A  C_  ( aleph `  A ) ) )
7 alephord2i 8490 . . . . . 6  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( aleph `  y )  e.  ( aleph `  x )
) )
87imp 427 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( aleph `  y )  e.  ( aleph `  x )
)
9 onelon 5435 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
10 alephon 8482 . . . . . 6  |-  ( aleph `  x )  e.  On
11 ontr2 5457 . . . . . 6  |-  ( ( y  e.  On  /\  ( aleph `  x )  e.  On )  ->  (
( y  C_  ( aleph `  y )  /\  ( aleph `  y )  e.  ( aleph `  x )
)  ->  y  e.  ( aleph `  x )
) )
129, 10, 11sylancl 660 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( ( y  C_  ( aleph `  y )  /\  ( aleph `  y )  e.  ( aleph `  x )
)  ->  y  e.  ( aleph `  x )
) )
138, 12mpan2d 672 . . . 4  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( y  C_  ( aleph `  y )  -> 
y  e.  ( aleph `  x ) ) )
1413ralimdva 2812 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  y  C_  ( aleph `  y
)  ->  A. y  e.  x  y  e.  ( aleph `  x )
) )
1510onirri 5516 . . . . 5  |-  -.  ( aleph `  x )  e.  ( aleph `  x )
16 eleq1 2474 . . . . . 6  |-  ( y  =  ( aleph `  x
)  ->  ( y  e.  ( aleph `  x )  <->  (
aleph `  x )  e.  ( aleph `  x )
) )
1716rspccv 3157 . . . . 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 5444 . . . . 5  |-  ( ( x  e.  On  /\  ( aleph `  x )  e.  On )  ->  (
x  C_  ( aleph `  x )  <->  -.  ( aleph `  x )  e.  x ) )
2010, 19mpan2 669 . . . 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 42 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  y  C_  ( aleph `  y
)  ->  x  C_  ( aleph `  x ) ) )
233, 6, 22tfis3 6675 1  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754    C_ wss 3414   Oncon0 5410   ` cfv 5569   alephcale 8349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-oi 7969  df-har 8018  df-card 8352  df-aleph 8353
This theorem is referenced by:  cardaleph  8502  alephfp  8521  winafp  9105
  Copyright terms: Public domain W3C validator