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Theorem alephsuc3 8419
Description: An alternate representation of a successor aleph. Compare alephsuc 7913 and alephsuc2 7925. Equality can be obtained by taking the  card of the right-hand side then using alephcard 7915 and carden 8390. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
alephsuc3  |-  ( A  e.  On  ->  ( aleph `  suc  A ) 
~~  { x  e.  On  |  x  ~~  ( aleph `  A ) } )
Distinct variable group:    x, A

Proof of Theorem alephsuc3
StepHypRef Expression
1 alephsuc2 7925 . . . . 5  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
2 alephcard 7915 . . . . . . 7  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
3 alephon 7914 . . . . . . . . 9  |-  ( aleph `  A )  e.  On
4 onenon 7800 . . . . . . . . 9  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
53, 4ax-mp 8 . . . . . . . 8  |-  ( aleph `  A )  e.  dom  card
6 cardval2 7842 . . . . . . . 8  |-  ( (
aleph `  A )  e. 
dom  card  ->  ( card `  ( aleph `  A )
)  =  { x  e.  On  |  x  ~<  (
aleph `  A ) } )
75, 6ax-mp 8 . . . . . . 7  |-  ( card `  ( aleph `  A )
)  =  { x  e.  On  |  x  ~<  (
aleph `  A ) }
82, 7eqtr3i 2434 . . . . . 6  |-  ( aleph `  A )  =  {
x  e.  On  |  x  ~<  ( aleph `  A
) }
98a1i 11 . . . . 5  |-  ( A  e.  On  ->  ( aleph `  A )  =  { x  e.  On  |  x  ~<  ( aleph `  A ) } )
101, 9difeq12d 3434 . . . 4  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  =  ( { x  e.  On  |  x  ~<_  ( aleph `  A ) }  \  { x  e.  On  |  x  ~<  ( aleph `  A ) } ) )
11 difrab 3583 . . . . 5  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  ( x  ~<_  (
aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
12 bren2 7105 . . . . . . 7  |-  ( x 
~~  ( aleph `  A
)  <->  ( x  ~<_  (
aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) )
1312a1i 11 . . . . . 6  |-  ( x  e.  On  ->  (
x  ~~  ( aleph `  A )  <->  ( x  ~<_  ( aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) ) )
1413rabbiia 2914 . . . . 5  |-  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  { x  e.  On  |  ( x  ~<_  ( aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
1511, 14eqtr4i 2435 . . . 4  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  x  ~~  ( aleph `  A ) }
1610, 15syl6req 2461 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  ( ( aleph `  suc  A ) 
\  ( aleph `  A
) ) )
17 alephon 7914 . . . . 5  |-  ( aleph ` 
suc  A )  e.  On
18 onenon 7800 . . . . 5  |-  ( (
aleph `  suc  A )  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
1917, 18mp1i 12 . . . 4  |-  ( A  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
20 sucelon 4764 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
21 alephgeom 7927 . . . . . 6  |-  ( suc 
A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
2220, 21bitri 241 . . . . 5  |-  ( A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
23 fvex 5709 . . . . . 6  |-  ( aleph ` 
suc  A )  e. 
_V
24 ssdomg 7120 . . . . . 6  |-  ( (
aleph `  suc  A )  e.  _V  ->  ( om  C_  ( aleph `  suc  A )  ->  om  ~<_  ( aleph ` 
suc  A ) ) )
2523, 24ax-mp 8 . . . . 5  |-  ( om  C_  ( aleph `  suc  A )  ->  om  ~<_  ( aleph ` 
suc  A ) )
2622, 25sylbi 188 . . . 4  |-  ( A  e.  On  ->  om  ~<_  ( aleph ` 
suc  A ) )
27 alephordilem1 7918 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
28 infdif 8053 . . . 4  |-  ( ( ( aleph `  suc  A )  e.  dom  card  /\  om  ~<_  ( aleph `  suc  A )  /\  ( aleph `  A
)  ~<  ( aleph `  suc  A ) )  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  ~~  ( aleph `  suc  A ) )
2919, 26, 27, 28syl3anc 1184 . . 3  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  ~~  ( aleph `  suc  A ) )
3016, 29eqbrtrd 4200 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  ~~  ( aleph `  suc  A ) )
3130ensymd 7125 1  |-  ( A  e.  On  ->  ( aleph `  suc  A ) 
~~  { x  e.  On  |  x  ~~  ( aleph `  A ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2678   _Vcvv 2924    \ cdif 3285    C_ wss 3288   class class class wbr 4180   Oncon0 4549   suc csuc 4551   omcom 4812   dom cdm 4845   ` cfv 5421    ~~ cen 7073    ~<_ cdom 7074    ~< csdm 7075   cardccrd 7786   alephcale 7787
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-oi 7443  df-har 7490  df-card 7790  df-aleph 7791  df-cda 8012
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