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Theorem alephsuc3 8460
Description: An alternate representation of a successor aleph. Compare alephsuc 7954 and alephsuc2 7966. Equality can be obtained by taking the  card of the right-hand side then using alephcard 7956 and carden 8431. (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 7966 . . . . 5  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
2 alephcard 7956 . . . . . . 7  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
3 alephon 7955 . . . . . . . . 9  |-  ( aleph `  A )  e.  On
4 onenon 7841 . . . . . . . . 9  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
53, 4ax-mp 5 . . . . . . . 8  |-  ( aleph `  A )  e.  dom  card
6 cardval2 7883 . . . . . . . 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 2460 . . . . . 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 3468 . . . 4  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  =  ( { x  e.  On  |  x  ~<_  ( aleph `  A ) }  \  { x  e.  On  |  x  ~<  ( aleph `  A ) } ) )
11 difrab 3617 . . . . 5  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  ( x  ~<_  (
aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
12 bren2 7141 . . . . . . 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 2948 . . . . 5  |-  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  { x  e.  On  |  ( x  ~<_  ( aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
1511, 14eqtr4i 2461 . . . 4  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  x  ~~  ( aleph `  A ) }
1610, 15syl6req 2487 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  ( ( aleph `  suc  A ) 
\  ( aleph `  A
) ) )
17 alephon 7955 . . . . 5  |-  ( aleph ` 
suc  A )  e.  On
18 onenon 7841 . . . . 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 4800 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
21 alephgeom 7968 . . . . . 6  |-  ( suc 
A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
2220, 21bitri 242 . . . . 5  |-  ( A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
23 fvex 5745 . . . . . 6  |-  ( aleph ` 
suc  A )  e. 
_V
24 ssdomg 7156 . . . . . 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 189 . . . 4  |-  ( A  e.  On  ->  om  ~<_  ( aleph ` 
suc  A ) )
27 alephordilem1 7959 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
28 infdif 8094 . . . 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 1185 . . 3  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  ~~  ( aleph `  suc  A ) )
3016, 29eqbrtrd 4235 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  ~~  ( aleph `  suc  A ) )
3130ensymd 7161 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    \ cdif 3319    C_ wss 3322   class class class wbr 4215   Oncon0 4584   suc csuc 4586   omcom 4848   dom cdm 4881   ` cfv 5457    ~~ cen 7109    ~<_ cdom 7110    ~< csdm 7111   cardccrd 7827   alephcale 7828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-oi 7482  df-har 7529  df-card 7831  df-aleph 7832  df-cda 8053
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