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Theorem tskwe 8120
Description: A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
tskwe  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  A  e.  dom  card )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem tskwe
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4476 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 rabexg 4442 . . . 4  |-  ( ~P A  e.  _V  ->  { x  e.  ~P A  |  x  ~<  A }  e.  _V )
3 incom 3543 . . . . 5  |-  ( { x  e.  ~P A  |  x  ~<  A }  i^i  On )  =  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )
4 inex1g 4435 . . . . 5  |-  ( { x  e.  ~P A  |  x  ~<  A }  e.  _V  ->  ( {
x  e.  ~P A  |  x  ~<  A }  i^i  On )  e.  _V )
53, 4syl5eqelr 2528 . . . 4  |-  ( { x  e.  ~P A  |  x  ~<  A }  e.  _V  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } )  e.  _V )
6 inss1 3570 . . . . . . . . . . 11  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  On
76sseli 3352 . . . . . . . . . 10  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  On )
8 onelon 4744 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
98ancoms 453 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  On )  ->  y  e.  On )
107, 9sylan2 474 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  On )
11 onelss 4761 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
1211impcom 430 . . . . . . . . . . . . 13  |-  ( ( y  e.  z  /\  z  e.  On )  ->  y  C_  z )
137, 12sylan2 474 . . . . . . . . . . . 12  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  C_  z )
14 inss2 3571 . . . . . . . . . . . . . . . . 17  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  { x  e.  ~P A  |  x 
~<  A }
1514sseli 3352 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  { x  e.  ~P A  |  x  ~<  A }
)
16 breq1 4295 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
x  ~<  A  <->  z  ~<  A ) )
1716elrab 3117 . . . . . . . . . . . . . . . 16  |-  ( z  e.  { x  e. 
~P A  |  x 
~<  A }  <->  ( z  e.  ~P A  /\  z  ~<  A ) )
1815, 17sylib 196 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  ( z  e.  ~P A  /\  z  ~<  A ) )
1918simpld 459 . . . . . . . . . . . . . 14  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  ~P A )
2019elpwid 3870 . . . . . . . . . . . . 13  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  C_  A )
2120adantl 466 . . . . . . . . . . . 12  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
z  C_  A )
2213, 21sstrd 3366 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  C_  A )
23 selpw 3867 . . . . . . . . . . 11  |-  ( y  e.  ~P A  <->  y  C_  A )
2422, 23sylibr 212 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  ~P A
)
25 vex 2975 . . . . . . . . . . . 12  |-  z  e. 
_V
26 ssdomg 7355 . . . . . . . . . . . 12  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
2725, 13, 26mpsyl 63 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  ~<_  z )
2818simprd 463 . . . . . . . . . . . 12  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  ~<  A )
2928adantl 466 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
z  ~<  A )
30 domsdomtr 7446 . . . . . . . . . . 11  |-  ( ( y  ~<_  z  /\  z  ~<  A )  ->  y  ~<  A )
3127, 29, 30syl2anc 661 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  ~<  A )
32 breq1 4295 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
x  ~<  A  <->  y  ~<  A ) )
3332elrab 3117 . . . . . . . . . 10  |-  ( y  e.  { x  e. 
~P A  |  x 
~<  A }  <->  ( y  e.  ~P A  /\  y  ~<  A ) )
3424, 31, 33sylanbrc 664 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  { x  e.  ~P A  |  x 
~<  A } )
3510, 34elind 3540 . . . . . . . 8  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } ) )
3635gen2 1592 . . . . . . 7  |-  A. y A. z ( ( y  e.  z  /\  z  e.  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
) )  ->  y  e.  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
) )
37 dftr2 4387 . . . . . . 7  |-  ( Tr  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  <->  A. y A. z
( ( y  e.  z  /\  z  e.  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
) )  ->  y  e.  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
) ) )
3836, 37mpbir 209 . . . . . 6  |-  Tr  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )
39 ordon 6394 . . . . . 6  |-  Ord  On
40 trssord 4736 . . . . . 6  |-  ( ( Tr  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  /\  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  On  /\ 
Ord  On )  ->  Ord  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } ) )
4138, 6, 39, 40mp3an 1314 . . . . 5  |-  Ord  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )
42 elong 4727 . . . . 5  |-  ( ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  e. 
_V  ->  ( ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  e.  On  <->  Ord  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
) ) )
4341, 42mpbiri 233 . . . 4  |-  ( ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  e. 
_V  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  e.  On )
441, 2, 5, 434syl 21 . . 3  |-  ( A  e.  V  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e.  On )
4544adantr 465 . 2  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e.  On )
46 simpr 461 . . . . 5  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  { x  e.  ~P A  |  x 
~<  A }  C_  A
)
4714, 46syl5ss 3367 . . . 4  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  C_  A )
48 ssdomg 7355 . . . . 5  |-  ( A  e.  V  ->  (
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  C_  A  ->  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  ~<_  A ) )
4948adantr 465 . . . 4  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  (
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  C_  A  ->  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  ~<_  A ) )
5047, 49mpd 15 . . 3  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  ~<_  A )
51 ordirr 4737 . . . . 5  |-  ( Ord  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  ->  -.  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e.  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
) )
5241, 51mp1i 12 . . . 4  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  -.  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  e.  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
) )
53443ad2ant1 1009 . . . . . 6  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A  /\  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  ~<  A )  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  e.  On )
54 elpw2g 4455 . . . . . . . . . 10  |-  ( A  e.  V  ->  (
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  e.  ~P A  <->  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  C_  A ) )
5554adantr 465 . . . . . . . . 9  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  (
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  e.  ~P A  <->  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  C_  A ) )
5647, 55mpbird 232 . . . . . . . 8  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e. 
~P A )
57563adant3 1008 . . . . . . 7  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A  /\  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  ~<  A )  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  e.  ~P A
)
58 simp3 990 . . . . . . 7  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A  /\  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  ~<  A )  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ~<  A )
59 nfcv 2579 . . . . . . . . 9  |-  F/_ x On
60 nfrab1 2901 . . . . . . . . 9  |-  F/_ x { x  e.  ~P A  |  x  ~<  A }
6159, 60nfin 3557 . . . . . . . 8  |-  F/_ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)
62 nfcv 2579 . . . . . . . 8  |-  F/_ x ~P A
63 nfcv 2579 . . . . . . . . 9  |-  F/_ x  ~<
64 nfcv 2579 . . . . . . . . 9  |-  F/_ x A
6561, 63, 64nfbr 4336 . . . . . . . 8  |-  F/ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  ~<  A
66 breq1 4295 . . . . . . . 8  |-  ( x  =  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  ( x  ~<  A  <->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ~<  A ) )
6761, 62, 65, 66elrabf 3115 . . . . . . 7  |-  ( ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  e. 
{ x  e.  ~P A  |  x  ~<  A }  <->  ( ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  e.  ~P A  /\  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ~<  A ) )
6857, 58, 67sylanbrc 664 . . . . . 6  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A  /\  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  ~<  A )  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  e.  { x  e.  ~P A  |  x 
~<  A } )
6953, 68elind 3540 . . . . 5  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A  /\  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  ~<  A )  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  e.  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } ) )
70693expia 1189 . . . 4  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  (
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  ~<  A  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e.  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
) ) )
7152, 70mtod 177 . . 3  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  -.  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  ~<  A )
72 bren2 7340 . . 3  |-  ( ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  ~~  A 
<->  ( ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ~<_  A  /\  -.  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  ~<  A ) )
7350, 71, 72sylanbrc 664 . 2  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  ~~  A )
74 isnumi 8116 . 2  |-  ( ( ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  e.  On  /\  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  ~~  A )  ->  A  e.  dom  card )
7545, 73, 74syl2anc 661 1  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  A  e.  dom  card )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    e. wcel 1756   {crab 2719   _Vcvv 2972    i^i cin 3327    C_ wss 3328   ~Pcpw 3860   class class class wbr 4292   Tr wtr 4385   Ord word 4718   Oncon0 4719   dom cdm 4840    ~~ cen 7307    ~<_ cdom 7308    ~< csdm 7309   cardccrd 8105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-card 8109
This theorem is referenced by:  tskwe2  8940  grothac  8997
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