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Theorem alephle 7953
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 7974, 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 20 . . 3  |-  ( x  =  y  ->  x  =  y )
2 fveq2 5714 . . 3  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
31, 2sseq12d 3364 . 2  |-  ( x  =  y  ->  (
x  C_  ( aleph `  x )  <->  y  C_  ( aleph `  y )
) )
4 id 20 . . 3  |-  ( x  =  A  ->  x  =  A )
5 fveq2 5714 . . 3  |-  ( x  =  A  ->  ( aleph `  x )  =  ( aleph `  A )
)
64, 5sseq12d 3364 . 2  |-  ( x  =  A  ->  (
x  C_  ( aleph `  x )  <->  A  C_  ( aleph `  A ) ) )
7 alephord2i 7942 . . . . . 6  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( aleph `  y )  e.  ( aleph `  x )
) )
87imp 419 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( aleph `  y )  e.  ( aleph `  x )
)
9 onelon 4593 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
10 alephon 7934 . . . . . 6  |-  ( aleph `  x )  e.  On
11 ontr2 4615 . . . . . 6  |-  ( ( y  e.  On  /\  ( aleph `  x )  e.  On )  ->  (
( y  C_  ( aleph `  y )  /\  ( aleph `  y )  e.  ( aleph `  x )
)  ->  y  e.  ( aleph `  x )
) )
129, 10, 11sylancl 644 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( ( y  C_  ( aleph `  y )  /\  ( aleph `  y )  e.  ( aleph `  x )
)  ->  y  e.  ( aleph `  x )
) )
138, 12mpan2d 656 . . . 4  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( y  C_  ( aleph `  y )  -> 
y  e.  ( aleph `  x ) ) )
1413ralimdva 2771 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  y  C_  ( aleph `  y
)  ->  A. y  e.  x  y  e.  ( aleph `  x )
) )
1510onirri 4674 . . . . 5  |-  -.  ( aleph `  x )  e.  ( aleph `  x )
16 eleq1 2490 . . . . . 6  |-  ( y  =  ( aleph `  x
)  ->  ( y  e.  ( aleph `  x )  <->  (
aleph `  x )  e.  ( aleph `  x )
) )
1716rspccv 3036 . . . . 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 4602 . . . . 5  |-  ( ( x  e.  On  /\  ( aleph `  x )  e.  On )  ->  (
x  C_  ( aleph `  x )  <->  -.  ( aleph `  x )  e.  x ) )
2010, 19mpan2 653 . . . 4  |-  ( x  e.  On  ->  (
x  C_  ( aleph `  x )  <->  -.  ( aleph `  x )  e.  x ) )
2118, 20syl5ibr 213 . . 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 4823 1  |-  ( A  e.  On  ->  A  C_  ( aleph `  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2692    C_ wss 3307   Oncon0 4568   ` cfv 5440   alephcale 7807
This theorem is referenced by:  cardaleph  7954  alephfp  7973  winafp  8556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-riota 6535  df-recs 6619  df-rdg 6654  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-oi 7463  df-har 7510  df-card 7810  df-aleph 7811
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