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Theorem alephadd 8153
Description: The sum 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
alephadd  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  A )  u.  ( aleph `  B )
)

Proof of Theorem alephadd
StepHypRef Expression
1 ovex 5803 . . . 4  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  e.  _V
2 alephfnon 7646 . . . . . . . 8  |-  aleph  Fn  On
3 fndm 5267 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
42, 3ax-mp 10 . . . . . . 7  |-  dom  aleph  =  On
54eleq2i 2320 . . . . . 6  |-  ( A  e.  dom  aleph  <->  A  e.  On )
65notbii 289 . . . . 5  |-  ( -.  A  e.  dom  aleph  <->  -.  A  e.  On )
74eleq2i 2320 . . . . . 6  |-  ( B  e.  dom  aleph  <->  B  e.  On )
87notbii 289 . . . . 5  |-  ( -.  B  e.  dom  aleph  <->  -.  B  e.  On )
9 0ex 4110 . . . . . . . 8  |-  (/)  e.  _V
10 cdaval 7750 . . . . . . . 8  |-  ( (
(/)  e.  _V  /\  (/)  e.  _V )  ->  ( (/)  +c  (/) )  =  ( ( (/)  X.  { (/)
} )  u.  ( (/) 
X.  { 1o }
) ) )
119, 9, 10mp2an 656 . . . . . . 7  |-  ( (/)  +c  (/) )  =  (
( (/)  X.  { (/) } )  u.  ( (/)  X. 
{ 1o } ) )
12 xpundi 4715 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  ( ( (/)  X. 
{ (/) } )  u.  ( (/)  X.  { 1o } ) )
13 xp0r 4742 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  (/)
1411, 12, 133eqtr2i 2282 . . . . . 6  |-  ( (/)  +c  (/) )  =  (/)
15 ndmfv 5472 . . . . . . 7  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
16 ndmfv 5472 . . . . . . 7  |-  ( -.  B  e.  dom  aleph  ->  ( aleph `  B )  =  (/) )
1715, 16oveqan12d 5797 . . . . . 6  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  +c  ( aleph `  B )
)  =  ( (/)  +c  (/) ) )
1815adantr 453 . . . . . . . 8  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( aleph `  A )  =  (/) )
1916adantl 454 . . . . . . . 8  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( aleph `  B )  =  (/) )
2018, 19uneq12d 3291 . . . . . . 7  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  ( (/)  u.  (/) ) )
21 un0 3440 . . . . . . 7  |-  ( (/)  u.  (/) )  =  (/)
2220, 21syl6eq 2304 . . . . . 6  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  (/) )
2314, 17, 223eqtr4a 2314 . . . . 5  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  +c  ( aleph `  B )
)  =  ( (
aleph `  A )  u.  ( aleph `  B )
) )
246, 8, 23syl2anbr 468 . . . 4  |-  ( ( -.  A  e.  On  /\ 
-.  B  e.  On )  ->  ( ( aleph `  A )  +c  ( aleph `  B ) )  =  ( ( aleph `  A )  u.  ( aleph `  B ) ) )
25 eqeng 6849 . . . 4  |-  ( ( ( aleph `  A )  +c  ( aleph `  B )
)  e.  _V  ->  ( ( ( aleph `  A
)  +c  ( aleph `  B ) )  =  ( ( aleph `  A
)  u.  ( aleph `  B ) )  -> 
( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) ) )
261, 24, 25mpsyl 61 . . 3  |-  ( ( -.  A  e.  On  /\ 
-.  B  e.  On )  ->  ( ( aleph `  A )  +c  ( aleph `  B ) ) 
~~  ( ( aleph `  A )  u.  ( aleph `  B ) ) )
2726ex 425 . 2  |-  ( -.  A  e.  On  ->  ( -.  B  e.  On  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) ) )
28 alephgeom 7663 . . 3  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
29 fvex 5458 . . . . 5  |-  ( aleph `  A )  e.  _V
30 ssdomg 6861 . . . . 5  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
3129, 30ax-mp 10 . . . 4  |-  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) )
32 alephon 7650 . . . . . 6  |-  ( aleph `  A )  e.  On
33 onenon 7536 . . . . . 6  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
3432, 33ax-mp 10 . . . . 5  |-  ( aleph `  A )  e.  dom  card
35 alephon 7650 . . . . . 6  |-  ( aleph `  B )  e.  On
36 onenon 7536 . . . . . 6  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
3735, 36ax-mp 10 . . . . 5  |-  ( aleph `  B )  e.  dom  card
38 infcda 7788 . . . . 5  |-  ( ( ( aleph `  A )  e.  dom  card  /\  ( aleph `  B )  e. 
dom  card  /\  om  ~<_  ( aleph `  A ) )  -> 
( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
3934, 37, 38mp3an12 1272 . . . 4  |-  ( om  ~<_  ( aleph `  A )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
4031, 39syl 17 . . 3  |-  ( om  C_  ( aleph `  A )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
4128, 40sylbi 189 . 2  |-  ( A  e.  On  ->  (
( aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  A )  u.  ( aleph `  B )
) )
42 alephgeom 7663 . . 3  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
43 fvex 5458 . . . . 5  |-  ( aleph `  B )  e.  _V
44 ssdomg 6861 . . . . 5  |-  ( (
aleph `  B )  e. 
_V  ->  ( om  C_  ( aleph `  B )  ->  om 
~<_  ( aleph `  B )
) )
4543, 44ax-mp 10 . . . 4  |-  ( om  C_  ( aleph `  B )  ->  om  ~<_  ( aleph `  B
) )
46 cdacomen 7761 . . . . . 6  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  B )  +c  ( aleph `  A )
)
47 infcda 7788 . . . . . . 7  |-  ( ( ( aleph `  B )  e.  dom  card  /\  ( aleph `  A )  e. 
dom  card  /\  om  ~<_  ( aleph `  B ) )  -> 
( ( aleph `  B
)  +c  ( aleph `  A ) )  ~~  ( ( aleph `  B
)  u.  ( aleph `  A ) ) )
4837, 34, 47mp3an12 1272 . . . . . 6  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  B
)  +c  ( aleph `  A ) )  ~~  ( ( aleph `  B
)  u.  ( aleph `  A ) ) )
49 entr 6867 . . . . . 6  |-  ( ( ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  B
)  +c  ( aleph `  A ) )  /\  ( ( aleph `  B
)  +c  ( aleph `  A ) )  ~~  ( ( aleph `  B
)  u.  ( aleph `  A ) ) )  ->  ( ( aleph `  A )  +c  ( aleph `  B ) ) 
~~  ( ( aleph `  B )  u.  ( aleph `  A ) ) )
5046, 48, 49sylancr 647 . . . . 5  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  B
)  u.  ( aleph `  A ) ) )
51 uncom 3280 . . . . 5  |-  ( (
aleph `  B )  u.  ( aleph `  A )
)  =  ( (
aleph `  A )  u.  ( aleph `  B )
)
5250, 51syl6breq 4022 . . . 4  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
5345, 52syl 17 . . 3  |-  ( om  C_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
5442, 53sylbi 189 . 2  |-  ( B  e.  On  ->  (
( aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  A )  u.  ( aleph `  B )
) )
5527, 41, 54pm2.61ii 159 1  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  A )  u.  ( aleph `  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2757    u. cun 3111    C_ wss 3113   (/)c0 3416   {csn 3600   class class class wbr 3983   Oncon0 4350   omcom 4614    X. cxp 4645   dom cdm 4647    Fn wfn 4654   ` cfv 4659  (class class class)co 5778   1oc1o 6426    ~~ cen 6814    ~<_ cdom 6815   cardccrd 7522   alephcale 7523    +c ccda 7747
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 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296
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 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-oi 7179  df-har 7226  df-card 7526  df-aleph 7527  df-cda 7748
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