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Theorem alephmul 8846
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 8356 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
2 fvex 5802 . . . . 5  |-  ( aleph `  A )  e.  _V
3 ssdomg 7458 . . . . 5  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
42, 3ax-mp 5 . . . 4  |-  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) )
51, 4sylbi 195 . . 3  |-  ( A  e.  On  ->  om  ~<_  ( aleph `  A ) )
6 alephon 8343 . . . 4  |-  ( aleph `  A )  e.  On
7 onenon 8223 . . . 4  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
86, 7ax-mp 5 . . 3  |-  ( aleph `  A )  e.  dom  card
95, 8jctil 537 . 2  |-  ( A  e.  On  ->  (
( aleph `  A )  e.  dom  card  /\  om  ~<_  ( aleph `  A ) ) )
10 alephgeom 8356 . . . 4  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
11 fvex 5802 . . . . . 6  |-  ( aleph `  B )  e.  _V
12 ssdomg 7458 . . . . . 6  |-  ( (
aleph `  B )  e. 
_V  ->  ( om  C_  ( aleph `  B )  ->  om 
~<_  ( aleph `  B )
) )
1311, 12ax-mp 5 . . . . 5  |-  ( om  C_  ( aleph `  B )  ->  om  ~<_  ( aleph `  B
) )
14 infn0 7678 . . . . 5  |-  ( om  ~<_  ( aleph `  B )  ->  ( aleph `  B )  =/=  (/) )
1513, 14syl 16 . . . 4  |-  ( om  C_  ( aleph `  B )  ->  ( aleph `  B )  =/=  (/) )
1610, 15sylbi 195 . . 3  |-  ( B  e.  On  ->  ( aleph `  B )  =/=  (/) )
17 alephon 8343 . . . 4  |-  ( aleph `  B )  e.  On
18 onenon 8223 . . . 4  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
1917, 18ax-mp 5 . . 3  |-  ( aleph `  B )  e.  dom  card
2016, 19jctil 537 . 2  |-  ( B  e.  On  ->  (
( aleph `  B )  e.  dom  card  /\  ( aleph `  B )  =/=  (/) ) )
21 infxp 8488 . 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 477 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  X.  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758    =/= wne 2644   _Vcvv 3071    u. cun 3427    C_ wss 3429   (/)c0 3738   class class class wbr 4393   Oncon0 4820    X. cxp 4939   dom cdm 4941   ` cfv 5519   omcom 6579    ~~ cen 7410    ~<_ cdom 7411   cardccrd 8209   alephcale 8210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-oi 7828  df-har 7877  df-card 8213  df-aleph 8214  df-cda 8441
This theorem is referenced by: (None)
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