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Theorem alephsuc3 8204
Description: An alternate representation of a successor aleph. Compare alephsuc 7697 and alephsuc2 7709. Equality can be obtained by taking the  card of the right-hand side then using alephcard 7699 and carden 8175. (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 7709 . . . . 5  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
2 alephcard 7699 . . . . . . 7  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
3 alephon 7698 . . . . . . . . 9  |-  ( aleph `  A )  e.  On
4 onenon 7584 . . . . . . . . 9  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
53, 4ax-mp 8 . . . . . . . 8  |-  ( aleph `  A )  e.  dom  card
6 cardval2 7626 . . . . . . . 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 2307 . . . . . 6  |-  ( aleph `  A )  =  {
x  e.  On  |  x  ~<  ( aleph `  A
) }
98a1i 10 . . . . 5  |-  ( A  e.  On  ->  ( aleph `  A )  =  { x  e.  On  |  x  ~<  ( aleph `  A ) } )
101, 9difeq12d 3297 . . . 4  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  =  ( { x  e.  On  |  x  ~<_  ( aleph `  A ) }  \  { x  e.  On  |  x  ~<  ( aleph `  A ) } ) )
11 difrab 3444 . . . . 5  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  ( x  ~<_  (
aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
12 bren2 6894 . . . . . . 7  |-  ( x 
~~  ( aleph `  A
)  <->  ( x  ~<_  (
aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) )
1312a1i 10 . . . . . 6  |-  ( x  e.  On  ->  (
x  ~~  ( aleph `  A )  <->  ( x  ~<_  ( aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) ) )
1413rabbiia 2780 . . . . 5  |-  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  { x  e.  On  |  ( x  ~<_  ( aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
1511, 14eqtr4i 2308 . . . 4  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  x  ~~  ( aleph `  A ) }
1610, 15syl6req 2334 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  ( ( aleph `  suc  A ) 
\  ( aleph `  A
) ) )
17 alephon 7698 . . . . 5  |-  ( aleph ` 
suc  A )  e.  On
18 onenon 7584 . . . . 5  |-  ( (
aleph `  suc  A )  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
1917, 18mp1i 11 . . . 4  |-  ( A  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
20 sucelon 4610 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
21 alephgeom 7711 . . . . . 6  |-  ( suc 
A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
2220, 21bitri 240 . . . . 5  |-  ( A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
23 fvex 5541 . . . . . 6  |-  ( aleph ` 
suc  A )  e. 
_V
24 ssdomg 6909 . . . . . 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 187 . . . 4  |-  ( A  e.  On  ->  om  ~<_  ( aleph ` 
suc  A ) )
27 alephordilem1 7702 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
28 infdif 7837 . . . 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 1182 . . 3  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  ~~  ( aleph `  suc  A ) )
3016, 29eqbrtrd 4045 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  ~~  ( aleph `  suc  A ) )
31 ensym 6912 . 2  |-  ( { x  e.  On  |  x  ~~  ( aleph `  A
) }  ~~  ( aleph `  suc  A )  ->  ( aleph `  suc  A )  ~~  { x  e.  On  |  x  ~~  ( aleph `  A ) } )
3230, 31syl 15 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 176    /\ wa 358    = wceq 1625    e. wcel 1686   {crab 2549   _Vcvv 2790    \ cdif 3151    C_ wss 3154   class class class wbr 4025   Oncon0 4394   suc csuc 4396   omcom 4658   dom cdm 4691   ` cfv 5257    ~~ cen 6862    ~<_ cdom 6863    ~< csdm 6864   cardccrd 7570   alephcale 7571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-oi 7227  df-har 7274  df-card 7574  df-aleph 7575  df-cda 7796
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