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Theorem alephadd 8408
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 6065 . . . 4  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  e.  _V
2 alephfnon 7902 . . . . . . . 8  |-  aleph  Fn  On
3 fndm 5503 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
42, 3ax-mp 8 . . . . . . 7  |-  dom  aleph  =  On
54eleq2i 2468 . . . . . 6  |-  ( A  e.  dom  aleph  <->  A  e.  On )
65notbii 288 . . . . 5  |-  ( -.  A  e.  dom  aleph  <->  -.  A  e.  On )
74eleq2i 2468 . . . . . 6  |-  ( B  e.  dom  aleph  <->  B  e.  On )
87notbii 288 . . . . 5  |-  ( -.  B  e.  dom  aleph  <->  -.  B  e.  On )
9 0ex 4299 . . . . . . . 8  |-  (/)  e.  _V
10 cdaval 8006 . . . . . . . 8  |-  ( (
(/)  e.  _V  /\  (/)  e.  _V )  ->  ( (/)  +c  (/) )  =  ( ( (/)  X.  { (/)
} )  u.  ( (/) 
X.  { 1o }
) ) )
119, 9, 10mp2an 654 . . . . . . 7  |-  ( (/)  +c  (/) )  =  (
( (/)  X.  { (/) } )  u.  ( (/)  X. 
{ 1o } ) )
12 xpundi 4889 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  ( ( (/)  X. 
{ (/) } )  u.  ( (/)  X.  { 1o } ) )
13 xp0r 4915 . . . . . . 7  |-  ( (/)  X.  ( { (/) }  u.  { 1o } ) )  =  (/)
1411, 12, 133eqtr2i 2430 . . . . . 6  |-  ( (/)  +c  (/) )  =  (/)
15 ndmfv 5714 . . . . . . 7  |-  ( -.  A  e.  dom  aleph  ->  ( aleph `  A )  =  (/) )
16 ndmfv 5714 . . . . . . 7  |-  ( -.  B  e.  dom  aleph  ->  ( aleph `  B )  =  (/) )
1715, 16oveqan12d 6059 . . . . . 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 3462 . . . . . . 7  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  ( (/)  u.  (/) ) )
21 un0 3612 . . . . . . 7  |-  ( (/)  u.  (/) )  =  (/)
2220, 21syl6eq 2452 . . . . . 6  |-  ( ( -.  A  e.  dom  aleph  /\  -.  B  e.  dom  aleph
)  ->  ( ( aleph `  A )  u.  ( aleph `  B )
)  =  (/) )
2314, 17, 223eqtr4a 2462 . . . . 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 7100 . . . 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 7919 . . 3  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
29 fvex 5701 . . . . 5  |-  ( aleph `  A )  e.  _V
30 ssdomg 7112 . . . . 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 7906 . . . . . 6  |-  ( aleph `  A )  e.  On
33 onenon 7792 . . . . . 6  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
3432, 33ax-mp 8 . . . . 5  |-  ( aleph `  A )  e.  dom  card
35 alephon 7906 . . . . . 6  |-  ( aleph `  B )  e.  On
36 onenon 7792 . . . . . 6  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
3735, 36ax-mp 8 . . . . 5  |-  ( aleph `  B )  e.  dom  card
38 infcda 8044 . . . . 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 7919 . . 3  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
43 fvex 5701 . . . . 5  |-  ( aleph `  B )  e.  _V
44 ssdomg 7112 . . . . 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 8017 . . . . . 6  |-  ( (
aleph `  A )  +c  ( aleph `  B )
)  ~~  ( ( aleph `  B )  +c  ( aleph `  A )
)
47 infcda 8044 . . . . . . 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 7118 . . . . . 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 3451 . . . . 5  |-  ( (
aleph `  B )  u.  ( aleph `  A )
)  =  ( (
aleph `  A )  u.  ( aleph `  B )
)
5250, 51syl6breq 4211 . . . 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 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172   Oncon0 4541   omcom 4804    X. cxp 4835   dom cdm 4837    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    ~<_ cdom 7066   cardccrd 7778   alephcale 7779    +c ccda 8003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783  df-cda 8004
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