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Theorem alephexp1 8943
Description: An exponentiation law for alephs. Lemma 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
alephexp1  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ( 2o  ^m  ( aleph `  B )
) )

Proof of Theorem alephexp1
StepHypRef Expression
1 alephon 8439 . . . 4  |-  ( aleph `  B )  e.  On
2 onenon 8319 . . . 4  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
31, 2mp1i 12 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  B )  e.  dom  card )
4 fvex 5867 . . . 4  |-  ( aleph `  B )  e.  _V
5 simplr 754 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  B  e.  On )
6 alephgeom 8452 . . . . 5  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
75, 6sylib 196 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  C_  ( aleph `  B ) )
8 ssdomg 7551 . . . 4  |-  ( (
aleph `  B )  e. 
_V  ->  ( om  C_  ( aleph `  B )  ->  om 
~<_  ( aleph `  B )
) )
94, 7, 8mpsyl 63 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  ~<_  ( aleph `  B ) )
10 fvex 5867 . . . 4  |-  ( aleph `  A )  e.  _V
11 ordom 6680 . . . . . 6  |-  Ord  om
12 2onn 7279 . . . . . 6  |-  2o  e.  om
13 ordelss 4887 . . . . . 6  |-  ( ( Ord  om  /\  2o  e.  om )  ->  2o  C_ 
om )
1411, 12, 13mp2an 672 . . . . 5  |-  2o  C_  om
15 simpll 753 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  A  e.  On )
16 alephgeom 8452 . . . . . 6  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
1715, 16sylib 196 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  C_  ( aleph `  A ) )
1814, 17syl5ss 3508 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  2o  C_  ( aleph `  A ) )
19 ssdomg 7551 . . . 4  |-  ( (
aleph `  A )  e. 
_V  ->  ( 2o  C_  ( aleph `  A )  ->  2o  ~<_  ( aleph `  A
) ) )
2010, 18, 19mpsyl 63 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  2o  ~<_  ( aleph `  A ) )
21 alephord3 8448 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  (
aleph `  A )  C_  ( aleph `  B )
) )
22 ssdomg 7551 . . . . . . 7  |-  ( (
aleph `  B )  e. 
_V  ->  ( ( aleph `  A )  C_  ( aleph `  B )  -> 
( aleph `  A )  ~<_  ( aleph `  B )
) )
234, 22ax-mp 5 . . . . . 6  |-  ( (
aleph `  A )  C_  ( aleph `  B )  ->  ( aleph `  A )  ~<_  ( aleph `  B )
)
2421, 23syl6bi 228 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
2524imp 429 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  A )  ~<_  ( aleph `  B ) )
264canth2 7660 . . . . 5  |-  ( aleph `  B )  ~<  ~P ( aleph `  B )
27 sdomdom 7533 . . . . 5  |-  ( (
aleph `  B )  ~<  ~P ( aleph `  B )  ->  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)
2826, 27ax-mp 5 . . . 4  |-  ( aleph `  B )  ~<_  ~P ( aleph `  B )
29 domtr 7558 . . . 4  |-  ( ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
3025, 28, 29sylancl 662 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
31 mappwen 8482 . . 3  |-  ( ( ( ( aleph `  B
)  e.  dom  card  /\ 
om  ~<_  ( aleph `  B
) )  /\  ( 2o 
~<_  ( aleph `  A )  /\  ( aleph `  A )  ~<_  ~P ( aleph `  B )
) )  ->  (
( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B ) )
323, 9, 20, 30, 31syl22anc 1224 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B ) )
334pw2en 7614 . . 3  |-  ~P ( aleph `  B )  ~~  ( 2o  ^m  ( aleph `  B ) )
34 enen2 7648 . . 3  |-  ( ~P ( aleph `  B )  ~~  ( 2o  ^m  ( aleph `  B ) )  ->  ( ( (
aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B )  <->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ( 2o  ^m  ( aleph `  B )
) ) )
3533, 34ax-mp 5 . 2  |-  ( ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B )  <->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ( 2o  ^m  ( aleph `  B )
) )
3632, 35sylib 196 1  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ( 2o  ^m  ( aleph `  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1762   _Vcvv 3106    C_ wss 3469   ~Pcpw 4003   class class class wbr 4440   Ord word 4870   Oncon0 4871   dom cdm 4992   ` cfv 5579  (class class class)co 6275   omcom 6671   2oc2o 7114    ^m cmap 7410    ~~ cen 7503    ~<_ cdom 7504    ~< csdm 7505   cardccrd 8305   alephcale 8306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-oi 7924  df-har 7973  df-card 8309  df-aleph 8310
This theorem is referenced by:  alephexp2  8945
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