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Theorem alephmul 8195
Description: The product of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
alephmul  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  X.  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )

Proof of Theorem alephmul
StepHypRef Expression
1 alephgeom 7704 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
2 fvex 5499 . . . . 5  |-  ( aleph `  A )  e.  _V
3 ssdomg 6902 . . . . 5  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
42, 3ax-mp 10 . . . 4  |-  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) )
51, 4sylbi 189 . . 3  |-  ( A  e.  On  ->  om  ~<_  ( aleph `  A ) )
6 alephon 7691 . . . 4  |-  ( aleph `  A )  e.  On
7 onenon 7577 . . . 4  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
86, 7ax-mp 10 . . 3  |-  ( aleph `  A )  e.  dom  card
95, 8jctil 525 . 2  |-  ( A  e.  On  ->  (
( aleph `  A )  e.  dom  card  /\  om  ~<_  ( aleph `  A ) ) )
10 alephgeom 7704 . . . 4  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
11 fvex 5499 . . . . . 6  |-  ( aleph `  B )  e.  _V
12 ssdomg 6902 . . . . . 6  |-  ( (
aleph `  B )  e. 
_V  ->  ( om  C_  ( aleph `  B )  ->  om 
~<_  ( aleph `  B )
) )
1311, 12ax-mp 10 . . . . 5  |-  ( om  C_  ( aleph `  B )  ->  om  ~<_  ( aleph `  B
) )
14 infn0 7114 . . . . 5  |-  ( om  ~<_  ( aleph `  B )  ->  ( aleph `  B )  =/=  (/) )
1513, 14syl 17 . . . 4  |-  ( om  C_  ( aleph `  B )  ->  ( aleph `  B )  =/=  (/) )
1610, 15sylbi 189 . . 3  |-  ( B  e.  On  ->  ( aleph `  B )  =/=  (/) )
17 alephon 7691 . . . 4  |-  ( aleph `  B )  e.  On
18 onenon 7577 . . . 4  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
1917, 18ax-mp 10 . . 3  |-  ( aleph `  B )  e.  dom  card
2016, 19jctil 525 . 2  |-  ( B  e.  On  ->  (
( aleph `  B )  e.  dom  card  /\  ( aleph `  B )  =/=  (/) ) )
21 infxp 7836 . 2  |-  ( ( ( ( aleph `  A
)  e.  dom  card  /\ 
om  ~<_  ( aleph `  A
) )  /\  (
( aleph `  B )  e.  dom  card  /\  ( aleph `  B )  =/=  (/) ) )  ->  (
( aleph `  A )  X.  ( aleph `  B )
)  ~~  ( ( aleph `  A )  u.  ( aleph `  B )
) )
229, 20, 21syl2an 465 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  X.  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1688    =/= wne 2447   _Vcvv 2789    u. cun 3151    C_ wss 3153   (/)c0 3456   class class class wbr 4024   Oncon0 4391   omcom 4655    X. cxp 4686   dom cdm 4688   ` cfv 5221    ~~ cen 6855    ~<_ cdom 6856   cardccrd 7563   alephcale 7564
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-oi 7220  df-har 7267  df-card 7567  df-aleph 7568  df-cda 7789
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