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Theorem alephsuc3 9005
Description: An alternate representation of a successor aleph. Compare alephsuc 8499 and alephsuc2 8511. Equality can be obtained by taking the  card of the right-hand side then using alephcard 8501 and carden 8976. (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 8511 . . . . 5  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e.  On  |  x  ~<_  (
aleph `  A ) } )
2 alephcard 8501 . . . . . . 7  |-  ( card `  ( aleph `  A )
)  =  ( aleph `  A )
3 alephon 8500 . . . . . . . . 9  |-  ( aleph `  A )  e.  On
4 onenon 8383 . . . . . . . . 9  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
53, 4ax-mp 5 . . . . . . . 8  |-  ( aleph `  A )  e.  dom  card
6 cardval2 8425 . . . . . . . 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 2475 . . . . . 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 3552 . . . 4  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  =  ( { x  e.  On  |  x  ~<_  ( aleph `  A ) }  \  { x  e.  On  |  x  ~<  ( aleph `  A ) } ) )
11 difrab 3717 . . . . 5  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  ( x  ~<_  (
aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
12 bren2 7600 . . . . . . 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 3033 . . . . 5  |-  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  { x  e.  On  |  ( x  ~<_  ( aleph `  A )  /\  -.  x  ~<  ( aleph `  A ) ) }
1511, 14eqtr4i 2476 . . . 4  |-  ( { x  e.  On  |  x  ~<_  ( aleph `  A
) }  \  {
x  e.  On  |  x  ~<  ( aleph `  A
) } )  =  { x  e.  On  |  x  ~~  ( aleph `  A ) }
1610, 15syl6req 2502 . . 3  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  =  ( ( aleph `  suc  A ) 
\  ( aleph `  A
) ) )
17 alephon 8500 . . . . 5  |-  ( aleph ` 
suc  A )  e.  On
18 onenon 8383 . . . . 5  |-  ( (
aleph `  suc  A )  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
1917, 18mp1i 13 . . . 4  |-  ( A  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
20 sucelon 6644 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
21 alephgeom 8513 . . . . . 6  |-  ( suc 
A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
2220, 21bitri 253 . . . . 5  |-  ( A  e.  On  <->  om  C_  ( aleph `  suc  A ) )
23 fvex 5875 . . . . . 6  |-  ( aleph ` 
suc  A )  e. 
_V
24 ssdomg 7615 . . . . . 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 199 . . . 4  |-  ( A  e.  On  ->  om  ~<_  ( aleph ` 
suc  A ) )
27 alephordilem1 8504 . . . 4  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
28 infdif 8639 . . . 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 1268 . . 3  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
\  ( aleph `  A
) )  ~~  ( aleph `  suc  A ) )
3016, 29eqbrtrd 4423 . 2  |-  ( A  e.  On  ->  { x  e.  On  |  x  ~~  ( aleph `  A ) }  ~~  ( aleph `  suc  A ) )
3130ensymd 7620 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   {crab 2741   _Vcvv 3045    \ cdif 3401    C_ wss 3404   class class class wbr 4402   dom cdm 4834   Oncon0 5423   suc csuc 5425   ` cfv 5582   omcom 6692    ~~ cen 7566    ~<_ cdom 7567    ~< csdm 7568   cardccrd 8369   alephcale 8370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-oi 8025  df-har 8073  df-card 8373  df-aleph 8374  df-cda 8598
This theorem is referenced by: (None)
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