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Theorem alephadd 8194
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 5844 . . . 4  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  e.  _V
2 alephfnon 7687 . . . . . . . 8  |-  aleph  Fn  On
3 fndm 5308 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
42, 3ax-mp 10 . . . . . . 7  |-  dom  aleph  =  On
54eleq2i 2348 . . . . . 6  |-  ( A  e.  dom  aleph  <->  A  e.  On )
65notbii 289 . . . . 5  |-  ( -.  A  e.  dom  aleph  <->  -.  A  e.  On )
74eleq2i 2348 . . . . . 6  |-  ( B  e.  dom  aleph  <->  B  e.  On )
87notbii 289 . . . . 5  |-  ( -.  B  e.  dom  aleph  <->  -.  B  e.  On )
9 0ex 4151 . . . . . . . 8  |-  (/)  e.  _V
10 cdaval 7791 . . . . . . . 8  |-  ( (
(/)  e.  _V  /\  (/)  e.  _V )  ->  ( (/)  +c  (/) )  =  ( ( (/)  X.  { (/)
} )  u.  ( (/) 
X.  { 1o }
) ) )
119, 9, 10mp2an 655 . . . . . . 7  |-  ( (/)  +c  (/) )  =  (
( (/)  X.  { (/) } )  u.  ( (/)  X. 
{ 1o } ) )
12 xpundi 4740 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  ( ( (/)  X. 
{ (/) } )  u.  ( (/)  X.  { 1o } ) )
13 xp0r 4767 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  (/)
1411, 12, 133eqtr2i 2310 . . . . . 6  |-  ( (/)  +c  (/) )  =  (/)
15 ndmfv 5513 . . . . . . 7  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
16 ndmfv 5513 . . . . . . 7  |-  ( -.  B  e.  dom  aleph  ->  ( aleph `  B )  =  (/) )
1715, 16oveqan12d 5838 . . . . . 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 3331 . . . . . . 7  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  ( (/)  u.  (/) ) )
21 un0 3480 . . . . . . 7  |-  ( (/)  u.  (/) )  =  (/)
2220, 21syl6eq 2332 . . . . . 6  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  (/) )
2314, 17, 223eqtr4a 2342 . . . . 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 6890 . . . 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 7704 . . 3  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
29 fvex 5499 . . . . 5  |-  ( aleph `  A )  e.  _V
30 ssdomg 6902 . . . . 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 7691 . . . . . 6  |-  ( aleph `  A )  e.  On
33 onenon 7577 . . . . . 6  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
3432, 33ax-mp 10 . . . . 5  |-  ( aleph `  A )  e.  dom  card
35 alephon 7691 . . . . . 6  |-  ( aleph `  B )  e.  On
36 onenon 7577 . . . . . 6  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
3735, 36ax-mp 10 . . . . 5  |-  ( aleph `  B )  e.  dom  card
38 infcda 7829 . . . . 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 1269 . . . 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 7704 . . 3  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
43 fvex 5499 . . . . 5  |-  ( aleph `  B )  e.  _V
44 ssdomg 6902 . . . . 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 7802 . . . . . 6  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  B )  +c  ( aleph `  A )
)
47 infcda 7829 . . . . . . 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 1269 . . . . . 6  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  B
)  +c  ( aleph `  A ) )  ~~  ( ( aleph `  B
)  u.  ( aleph `  A ) ) )
49 entr 6908 . . . . . 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 646 . . . . 5  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  B
)  u.  ( aleph `  A ) ) )
51 uncom 3320 . . . . 5  |-  ( (
aleph `  B )  u.  ( aleph `  A )
)  =  ( (
aleph `  A )  u.  ( aleph `  B )
)
5250, 51syl6breq 4063 . . . 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 1624    e. wcel 1685   _Vcvv 2789    u. cun 3151    C_ wss 3153   (/)c0 3456   {csn 3641   class class class wbr 4024   Oncon0 4391   omcom 4655    X. cxp 4686   dom cdm 4688    Fn wfn 5216   ` cfv 5221  (class class class)co 5819   1oc1o 6467    ~~ cen 6855    ~<_ cdom 6856   cardccrd 7563   alephcale 7564    +c ccda 7788
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  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 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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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