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Theorem alephsuc3 8958
Description: An alternate representation of a successor aleph. Compare alephsuc 8452 and alephsuc2 8464. Equality can be obtained by taking the  card of the right-hand side then using alephcard 8454 and carden 8929. (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 8464 . . . . 5  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
2 alephcard 8454 . . . . . . 7  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
3 alephon 8453 . . . . . . . . 9  |-  ( aleph `  A )  e.  On
4 onenon 8333 . . . . . . . . 9  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
53, 4ax-mp 5 . . . . . . . 8  |-  ( aleph `  A )  e.  dom  card
6 cardval2 8375 . . . . . . . 8  |-  ( (
aleph `  A )  e. 
dom  card  ->  ( card `  ( aleph `  A )
)  =  { x  e.  On  |  x  ~<  (
aleph `  A ) } )
75, 6ax-mp 5 . . . . . . 7  |-  ( card `  ( aleph `  A )
)  =  { x  e.  On  |  x  ~<  (
aleph `  A ) }
82, 7eqtr3i 2474 . . . . . 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 3608 . . . 4  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  =  ( { x  e.  On  |  x  ~<_  ( aleph `  A ) }  \  { x  e.  On  |  x  ~<  ( aleph `  A ) } ) )
11 difrab 3757 . . . . 5  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  ( x  ~<_  (
aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
12 bren2 7548 . . . . . . 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 3084 . . . . 5  |-  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  { x  e.  On  |  ( x  ~<_  ( aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
1511, 14eqtr4i 2475 . . . 4  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  x  ~~  ( aleph `  A ) }
1610, 15syl6req 2501 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  ( ( aleph `  suc  A ) 
\  ( aleph `  A
) ) )
17 alephon 8453 . . . . 5  |-  ( aleph ` 
suc  A )  e.  On
18 onenon 8333 . . . . 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 6637 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
21 alephgeom 8466 . . . . . 6  |-  ( suc 
A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
2220, 21bitri 249 . . . . 5  |-  ( A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
23 fvex 5866 . . . . . 6  |-  ( aleph ` 
suc  A )  e. 
_V
24 ssdomg 7563 . . . . . 6  |-  ( (
aleph `  suc  A )  e.  _V  ->  ( om  C_  ( aleph `  suc  A )  ->  om  ~<_  ( aleph ` 
suc  A ) ) )
2523, 24ax-mp 5 . . . . 5  |-  ( om  C_  ( aleph `  suc  A )  ->  om  ~<_  ( aleph ` 
suc  A ) )
2622, 25sylbi 195 . . . 4  |-  ( A  e.  On  ->  om  ~<_  ( aleph ` 
suc  A ) )
27 alephordilem1 8457 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
28 infdif 8592 . . . 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 1229 . . 3  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  ~~  ( aleph `  suc  A ) )
3016, 29eqbrtrd 4457 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  ~~  ( aleph `  suc  A ) )
3130ensymd 7568 1  |-  ( A  e.  On  ->  ( aleph `  suc  A ) 
~~  { x  e.  On  |  x  ~~  ( aleph `  A ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   {crab 2797   _Vcvv 3095    \ cdif 3458    C_ wss 3461   class class class wbr 4437   Oncon0 4868   suc csuc 4870   dom cdm 4989   ` cfv 5578   omcom 6685    ~~ cen 7515    ~<_ cdom 7516    ~< csdm 7517   cardccrd 8319   alephcale 8320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-oi 7938  df-har 7987  df-card 8323  df-aleph 8324  df-cda 8551
This theorem is referenced by: (None)
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