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Theorem alephadd 8286
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 5967 . . . 4  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  e.  _V
2 alephfnon 7779 . . . . . . . 8  |-  aleph  Fn  On
3 fndm 5422 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
42, 3ax-mp 8 . . . . . . 7  |-  dom  aleph  =  On
54eleq2i 2422 . . . . . 6  |-  ( A  e.  dom  aleph  <->  A  e.  On )
65notbii 287 . . . . 5  |-  ( -.  A  e.  dom  aleph  <->  -.  A  e.  On )
74eleq2i 2422 . . . . . 6  |-  ( B  e.  dom  aleph  <->  B  e.  On )
87notbii 287 . . . . 5  |-  ( -.  B  e.  dom  aleph  <->  -.  B  e.  On )
9 0ex 4229 . . . . . . . 8  |-  (/)  e.  _V
10 cdaval 7883 . . . . . . . 8  |-  ( (
(/)  e.  _V  /\  (/)  e.  _V )  ->  ( (/)  +c  (/) )  =  ( ( (/)  X.  { (/)
} )  u.  ( (/) 
X.  { 1o }
) ) )
119, 9, 10mp2an 653 . . . . . . 7  |-  ( (/)  +c  (/) )  =  (
( (/)  X.  { (/) } )  u.  ( (/)  X. 
{ 1o } ) )
12 xpundi 4820 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  ( ( (/)  X. 
{ (/) } )  u.  ( (/)  X.  { 1o } ) )
13 xp0r 4847 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  (/)
1411, 12, 133eqtr2i 2384 . . . . . 6  |-  ( (/)  +c  (/) )  =  (/)
15 ndmfv 5632 . . . . . . 7  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
16 ndmfv 5632 . . . . . . 7  |-  ( -.  B  e.  dom  aleph  ->  ( aleph `  B )  =  (/) )
1715, 16oveqan12d 5961 . . . . . 6  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  +c  ( aleph `  B )
)  =  ( (/)  +c  (/) ) )
1815adantr 451 . . . . . . . 8  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( aleph `  A )  =  (/) )
1916adantl 452 . . . . . . . 8  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( aleph `  B )  =  (/) )
2018, 19uneq12d 3406 . . . . . . 7  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  ( (/)  u.  (/) ) )
21 un0 3555 . . . . . . 7  |-  ( (/)  u.  (/) )  =  (/)
2220, 21syl6eq 2406 . . . . . 6  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  (/) )
2314, 17, 223eqtr4a 2416 . . . . 5  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  +c  ( aleph `  B )
)  =  ( (
aleph `  A )  u.  ( aleph `  B )
) )
246, 8, 23syl2anbr 466 . . . 4  |-  ( ( -.  A  e.  On  /\ 
-.  B  e.  On )  ->  ( ( aleph `  A )  +c  ( aleph `  B ) )  =  ( ( aleph `  A )  u.  ( aleph `  B ) ) )
25 eqeng 6980 . . . 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 59 . . 3  |-  ( ( -.  A  e.  On  /\ 
-.  B  e.  On )  ->  ( ( aleph `  A )  +c  ( aleph `  B ) ) 
~~  ( ( aleph `  A )  u.  ( aleph `  B ) ) )
2726ex 423 . 2  |-  ( -.  A  e.  On  ->  ( -.  B  e.  On  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) ) )
28 alephgeom 7796 . . 3  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
29 fvex 5619 . . . . 5  |-  ( aleph `  A )  e.  _V
30 ssdomg 6992 . . . . 5  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
3129, 30ax-mp 8 . . . 4  |-  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) )
32 alephon 7783 . . . . . 6  |-  ( aleph `  A )  e.  On
33 onenon 7669 . . . . . 6  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
3432, 33ax-mp 8 . . . . 5  |-  ( aleph `  A )  e.  dom  card
35 alephon 7783 . . . . . 6  |-  ( aleph `  B )  e.  On
36 onenon 7669 . . . . . 6  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
3735, 36ax-mp 8 . . . . 5  |-  ( aleph `  B )  e.  dom  card
38 infcda 7921 . . . . 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 1267 . . . 4  |-  ( om  ~<_  ( aleph `  A )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
4031, 39syl 15 . . 3  |-  ( om  C_  ( aleph `  A )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
4128, 40sylbi 187 . 2  |-  ( A  e.  On  ->  (
( aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  A )  u.  ( aleph `  B )
) )
42 alephgeom 7796 . . 3  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
43 fvex 5619 . . . . 5  |-  ( aleph `  B )  e.  _V
44 ssdomg 6992 . . . . 5  |-  ( (
aleph `  B )  e. 
_V  ->  ( om  C_  ( aleph `  B )  ->  om 
~<_  ( aleph `  B )
) )
4543, 44ax-mp 8 . . . 4  |-  ( om  C_  ( aleph `  B )  ->  om  ~<_  ( aleph `  B
) )
46 cdacomen 7894 . . . . . 6  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  B )  +c  ( aleph `  A )
)
47 infcda 7921 . . . . . . 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 1267 . . . . . 6  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  B
)  +c  ( aleph `  A ) )  ~~  ( ( aleph `  B
)  u.  ( aleph `  A ) ) )
49 entr 6998 . . . . . 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 644 . . . . 5  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  B
)  u.  ( aleph `  A ) ) )
51 uncom 3395 . . . . 5  |-  ( (
aleph `  B )  u.  ( aleph `  A )
)  =  ( (
aleph `  A )  u.  ( aleph `  B )
)
5250, 51syl6breq 4141 . . . 4  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
5345, 52syl 15 . . 3  |-  ( om  C_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
5442, 53sylbi 187 . 2  |-  ( B  e.  On  ->  (
( aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  A )  u.  ( aleph `  B )
) )
5527, 41, 54pm2.61ii 157 1  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  A )  u.  ( aleph `  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864    u. cun 3226    C_ wss 3228   (/)c0 3531   {csn 3716   class class class wbr 4102   Oncon0 4471   omcom 4735    X. cxp 4766   dom cdm 4768    Fn wfn 5329   ` cfv 5334  (class class class)co 5942   1oc1o 6556    ~~ cen 6945    ~<_ cdom 6946   cardccrd 7655   alephcale 7656    +c ccda 7880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-oi 7312  df-har 7359  df-card 7659  df-aleph 7660  df-cda 7881
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