MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephmul Unicode version

Theorem alephmul 8133
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 7642 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
2 fvex 5437 . . . . 5  |-  ( aleph `  A )  e.  _V
3 ssdomg 6840 . . . . 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 7629 . . . 4  |-  ( aleph `  A )  e.  On
7 onenon 7515 . . . 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 7642 . . . 4  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
11 fvex 5437 . . . . . 6  |-  ( aleph `  B )  e.  _V
12 ssdomg 6840 . . . . . 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 7052 . . . . 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 7629 . . . 4  |-  ( aleph `  B )  e.  On
18 onenon 7515 . . . 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 7774 . 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 1621    =/= wne 2419   _Vcvv 2740    u. cun 3092    C_ wss 3094   (/)c0 3397   class class class wbr 3963   Oncon0 4329   omcom 4593    X. cxp 4624   dom cdm 4626   ` cfv 4638    ~~ cen 6793    ~<_ cdom 6794   cardccrd 7501   alephcale 7502
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275
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 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-har 7205  df-card 7505  df-aleph 7506  df-cda 7727
  Copyright terms: Public domain W3C validator