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Theorem alephadd 8441
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 6097 . . . 4  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  e.  _V
2 alephfnon 7935 . . . . . . . 8  |-  aleph  Fn  On
3 fndm 5535 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
42, 3ax-mp 8 . . . . . . 7  |-  dom  aleph  =  On
54eleq2i 2499 . . . . . 6  |-  ( A  e.  dom  aleph  <->  A  e.  On )
65notbii 288 . . . . 5  |-  ( -.  A  e.  dom  aleph  <->  -.  A  e.  On )
74eleq2i 2499 . . . . . 6  |-  ( B  e.  dom  aleph  <->  B  e.  On )
87notbii 288 . . . . 5  |-  ( -.  B  e.  dom  aleph  <->  -.  B  e.  On )
9 0ex 4331 . . . . . . . 8  |-  (/)  e.  _V
10 cdaval 8039 . . . . . . . 8  |-  ( (
(/)  e.  _V  /\  (/)  e.  _V )  ->  ( (/)  +c  (/) )  =  ( ( (/)  X.  { (/)
} )  u.  ( (/) 
X.  { 1o }
) ) )
119, 9, 10mp2an 654 . . . . . . 7  |-  ( (/)  +c  (/) )  =  (
( (/)  X.  { (/) } )  u.  ( (/)  X. 
{ 1o } ) )
12 xpundi 4921 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  ( ( (/)  X. 
{ (/) } )  u.  ( (/)  X.  { 1o } ) )
13 xp0r 4947 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  (/)
1411, 12, 133eqtr2i 2461 . . . . . 6  |-  ( (/)  +c  (/) )  =  (/)
15 ndmfv 5746 . . . . . . 7  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
16 ndmfv 5746 . . . . . . 7  |-  ( -.  B  e.  dom  aleph  ->  ( aleph `  B )  =  (/) )
1715, 16oveqan12d 6091 . . . . . 6  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  +c  ( aleph `  B )
)  =  ( (/)  +c  (/) ) )
1815adantr 452 . . . . . . . 8  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( aleph `  A )  =  (/) )
1916adantl 453 . . . . . . . 8  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( aleph `  B )  =  (/) )
2018, 19uneq12d 3494 . . . . . . 7  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  ( (/)  u.  (/) ) )
21 un0 3644 . . . . . . 7  |-  ( (/)  u.  (/) )  =  (/)
2220, 21syl6eq 2483 . . . . . 6  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  (/) )
2314, 17, 223eqtr4a 2493 . . . . 5  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  +c  ( aleph `  B )
)  =  ( (
aleph `  A )  u.  ( aleph `  B )
) )
246, 8, 23syl2anbr 467 . . . 4  |-  ( ( -.  A  e.  On  /\ 
-.  B  e.  On )  ->  ( ( aleph `  A )  +c  ( aleph `  B ) )  =  ( ( aleph `  A )  u.  ( aleph `  B ) ) )
25 eqeng 7132 . . . 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 424 . 2  |-  ( -.  A  e.  On  ->  ( -.  B  e.  On  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) ) )
28 alephgeom 7952 . . 3  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
29 fvex 5733 . . . . 5  |-  ( aleph `  A )  e.  _V
30 ssdomg 7144 . . . . 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 7939 . . . . . 6  |-  ( aleph `  A )  e.  On
33 onenon 7825 . . . . . 6  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
3432, 33ax-mp 8 . . . . 5  |-  ( aleph `  A )  e.  dom  card
35 alephon 7939 . . . . . 6  |-  ( aleph `  B )  e.  On
36 onenon 7825 . . . . . 6  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
3735, 36ax-mp 8 . . . . 5  |-  ( aleph `  B )  e.  dom  card
38 infcda 8077 . . . . 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 16 . . 3  |-  ( om  C_  ( aleph `  A )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
4128, 40sylbi 188 . 2  |-  ( A  e.  On  ->  (
( aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  A )  u.  ( aleph `  B )
) )
42 alephgeom 7952 . . 3  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
43 fvex 5733 . . . . 5  |-  ( aleph `  B )  e.  _V
44 ssdomg 7144 . . . . 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 8050 . . . . . 6  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  B )  +c  ( aleph `  A )
)
47 infcda 8077 . . . . . . 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 7150 . . . . . 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 645 . . . . 5  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  B
)  u.  ( aleph `  A ) ) )
51 uncom 3483 . . . . 5  |-  ( (
aleph `  B )  u.  ( aleph `  A )
)  =  ( (
aleph `  A )  u.  ( aleph `  B )
)
5250, 51syl6breq 4243 . . . 4  |-  ( om  ~<_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
5345, 52syl 16 . . 3  |-  ( om  C_  ( aleph `  B )  ->  ( ( aleph `  A
)  +c  ( aleph `  B ) )  ~~  ( ( aleph `  A
)  u.  ( aleph `  B ) ) )
5442, 53sylbi 188 . 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 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204   Oncon0 4573   omcom 4836    X. cxp 4867   dom cdm 4869    Fn wfn 5440   ` cfv 5445  (class class class)co 6072   1oc1o 6708    ~~ cen 7097    ~<_ cdom 7098   cardccrd 7811   alephcale 7812    +c ccda 8036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-oi 7468  df-har 7515  df-card 7815  df-aleph 7816  df-cda 8037
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