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Theorem tskwe 7583
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 4194 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 rabexg 4164 . . . . 5  |-  ( ~P A  e.  _V  ->  { x  e.  ~P A  |  x  ~<  A }  e.  _V )
3 incom 3361 . . . . . 6  |-  ( { x  e.  ~P A  |  x  ~<  A }  i^i  On )  =  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )
4 inex1g 4157 . . . . . 6  |-  ( { x  e.  ~P A  |  x  ~<  A }  e.  _V  ->  ( {
x  e.  ~P A  |  x  ~<  A }  i^i  On )  e.  _V )
53, 4syl5eqelr 2368 . . . . 5  |-  ( { x  e.  ~P A  |  x  ~<  A }  e.  _V  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } )  e.  _V )
61, 2, 53syl 18 . . . 4  |-  ( A  e.  V  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e. 
_V )
7 inss1 3389 . . . . . . . . . . 11  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  On
87sseli 3176 . . . . . . . . . 10  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  On )
9 onelon 4417 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
109ancoms 439 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  On )  ->  y  e.  On )
118, 10sylan2 460 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  On )
12 onelss 4434 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
1312impcom 419 . . . . . . . . . . . . 13  |-  ( ( y  e.  z  /\  z  e.  On )  ->  y  C_  z )
148, 13sylan2 460 . . . . . . . . . . . 12  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  C_  z )
15 inss2 3390 . . . . . . . . . . . . . . . . 17  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  { x  e.  ~P A  |  x 
~<  A }
1615sseli 3176 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  { x  e.  ~P A  |  x  ~<  A }
)
17 breq1 4026 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
x  ~<  A  <->  z  ~<  A ) )
1817elrab 2923 . . . . . . . . . . . . . . . 16  |-  ( z  e.  { x  e. 
~P A  |  x 
~<  A }  <->  ( z  e.  ~P A  /\  z  ~<  A ) )
1916, 18sylib 188 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  ( z  e.  ~P A  /\  z  ~<  A ) )
2019simpld 445 . . . . . . . . . . . . . 14  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  ~P A )
21 elpwi 3633 . . . . . . . . . . . . . 14  |-  ( z  e.  ~P A  -> 
z  C_  A )
2220, 21syl 15 . . . . . . . . . . . . 13  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  C_  A )
2322adantl 452 . . . . . . . . . . . 12  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
z  C_  A )
2414, 23sstrd 3189 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  C_  A )
25 vex 2791 . . . . . . . . . . . 12  |-  y  e. 
_V
2625elpw 3631 . . . . . . . . . . 11  |-  ( y  e.  ~P A  <->  y  C_  A )
2724, 26sylibr 203 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  ~P A
)
28 vex 2791 . . . . . . . . . . . 12  |-  z  e. 
_V
29 ssdomg 6907 . . . . . . . . . . . 12  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
3028, 14, 29mpsyl 59 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  ~<_  z )
3119simprd 449 . . . . . . . . . . . 12  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  ~<  A )
3231adantl 452 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
z  ~<  A )
33 domsdomtr 6996 . . . . . . . . . . 11  |-  ( ( y  ~<_  z  /\  z  ~<  A )  ->  y  ~<  A )
3430, 32, 33syl2anc 642 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  ~<  A )
35 breq1 4026 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
x  ~<  A  <->  y  ~<  A ) )
3635elrab 2923 . . . . . . . . . 10  |-  ( y  e.  { x  e. 
~P A  |  x 
~<  A }  <->  ( y  e.  ~P A  /\  y  ~<  A ) )
3727, 34, 36sylanbrc 645 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  { x  e.  ~P A  |  x 
~<  A } )
38 elin 3358 . . . . . . . . 9  |-  ( y  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  <->  ( y  e.  On  /\  y  e. 
{ x  e.  ~P A  |  x  ~<  A } ) )
3911, 37, 38sylanbrc 645 . . . . . . . 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 } ) )
4039gen2 1534 . . . . . . 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 }
) )
41 dftr2 4115 . . . . . . 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 }
) ) )
4240, 41mpbir 200 . . . . . 6  |-  Tr  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )
43 ordon 4574 . . . . . 6  |-  Ord  On
44 trssord 4409 . . . . . 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 } ) )
4542, 7, 43, 44mp3an 1277 . . . . 5  |-  Ord  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )
46 elong 4400 . . . . 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 }
) ) )
4745, 46mpbiri 224 . . . 4  |-  ( ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  e. 
_V  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  e.  On )
486, 47syl 15 . . 3  |-  ( A  e.  V  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e.  On )
4948adantr 451 . 2  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e.  On )
50 simpr 447 . . . . 5  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  { x  e.  ~P A  |  x 
~<  A }  C_  A
)
5115, 50syl5ss 3190 . . . 4  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  C_  A )
52 ssdomg 6907 . . . . 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 ) )
5352adantr 451 . . . 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 ) )
5451, 53mpd 14 . . 3  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  ~<_  A )
55 ordirr 4410 . . . . 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 }
) )
5645, 55mp1i 11 . . . 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 }
) )
57483ad2ant1 976 . . . . . 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 )
58 elpw2g 4174 . . . . . . . . . 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 ) )
5958adantr 451 . . . . . . . . 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 ) )
6051, 59mpbird 223 . . . . . . . 8  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e. 
~P A )
61603adant3 975 . . . . . . 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
)
62 simp3 957 . . . . . . 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 )
63 nfcv 2419 . . . . . . . . 9  |-  F/_ x On
64 nfrab1 2720 . . . . . . . . 9  |-  F/_ x { x  e.  ~P A  |  x  ~<  A }
6563, 64nfin 3375 . . . . . . . 8  |-  F/_ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)
66 nfcv 2419 . . . . . . . 8  |-  F/_ x ~P A
67 nfcv 2419 . . . . . . . . 9  |-  F/_ x  ~<
68 nfcv 2419 . . . . . . . . 9  |-  F/_ x A
6965, 67, 68nfbr 4067 . . . . . . . 8  |-  F/ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  ~<  A
70 breq1 4026 . . . . . . . 8  |-  ( x  =  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  ( x  ~<  A  <->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ~<  A ) )
7165, 66, 69, 70elrabf 2922 . . . . . . 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 ) )
7261, 62, 71sylanbrc 645 . . . . . 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 } )
73 elin 3358 . . . . . 6  |-  ( ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  e.  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  <->  ( ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  e.  On  /\  ( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  e.  { x  e.  ~P A  |  x 
~<  A } ) )
7457, 72, 73sylanbrc 645 . . . . 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 } ) )
75743expia 1153 . . . 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 }
) ) )
7656, 75mtod 168 . . 3  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  -.  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  ~<  A )
77 bren2 6892 . . 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 ) )
7854, 76, 77sylanbrc 645 . 2  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  ~~  A )
79 isnumi 7579 . 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 )
8049, 78, 79syl2anc 642 1  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  A  e.  dom  card )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    e. wcel 1684   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   Tr wtr 4113   Ord word 4391   Oncon0 4392   dom cdm 4689    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568
This theorem is referenced by:  tskwe2  8395  grothac  8452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572
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