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Theorem alephmul 8409
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 7919 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
2 fvex 5701 . . . . 5  |-  ( aleph `  A )  e.  _V
3 ssdomg 7112 . . . . 5  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
42, 3ax-mp 8 . . . 4  |-  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) )
51, 4sylbi 188 . . 3  |-  ( A  e.  On  ->  om  ~<_  ( aleph `  A ) )
6 alephon 7906 . . . 4  |-  ( aleph `  A )  e.  On
7 onenon 7792 . . . 4  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
86, 7ax-mp 8 . . 3  |-  ( aleph `  A )  e.  dom  card
95, 8jctil 524 . 2  |-  ( A  e.  On  ->  (
( aleph `  A )  e.  dom  card  /\  om  ~<_  ( aleph `  A ) ) )
10 alephgeom 7919 . . . 4  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
11 fvex 5701 . . . . . 6  |-  ( aleph `  B )  e.  _V
12 ssdomg 7112 . . . . . 6  |-  ( (
aleph `  B )  e. 
_V  ->  ( om  C_  ( aleph `  B )  ->  om 
~<_  ( aleph `  B )
) )
1311, 12ax-mp 8 . . . . 5  |-  ( om  C_  ( aleph `  B )  ->  om  ~<_  ( aleph `  B
) )
14 infn0 7328 . . . . 5  |-  ( om  ~<_  ( aleph `  B )  ->  ( aleph `  B )  =/=  (/) )
1513, 14syl 16 . . . 4  |-  ( om  C_  ( aleph `  B )  ->  ( aleph `  B )  =/=  (/) )
1610, 15sylbi 188 . . 3  |-  ( B  e.  On  ->  ( aleph `  B )  =/=  (/) )
17 alephon 7906 . . . 4  |-  ( aleph `  B )  e.  On
18 onenon 7792 . . . 4  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
1917, 18ax-mp 8 . . 3  |-  ( aleph `  B )  e.  dom  card
2016, 19jctil 524 . 2  |-  ( B  e.  On  ->  (
( aleph `  B )  e.  dom  card  /\  ( aleph `  B )  =/=  (/) ) )
21 infxp 8051 . 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 464 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 4    /\ wa 359    e. wcel 1721    =/= wne 2567   _Vcvv 2916    u. cun 3278    C_ wss 3280   (/)c0 3588   class class class wbr 4172   Oncon0 4541   omcom 4804    X. cxp 4835   dom cdm 4837   ` cfv 5413    ~~ cen 7065    ~<_ cdom 7066   cardccrd 7778   alephcale 7779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783  df-cda 8004
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