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Theorem tskwe 7599
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 4210 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 rabexg 4180 . . . . 5  |-  ( ~P A  e.  _V  ->  { x  e.  ~P A  |  x  ~<  A }  e.  _V )
3 incom 3374 . . . . . 6  |-  ( { x  e.  ~P A  |  x  ~<  A }  i^i  On )  =  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )
4 inex1g 4173 . . . . . 6  |-  ( { x  e.  ~P A  |  x  ~<  A }  e.  _V  ->  ( {
x  e.  ~P A  |  x  ~<  A }  i^i  On )  e.  _V )
53, 4syl5eqelr 2381 . . . . 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 3402 . . . . . . . . . . 11  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  On
87sseli 3189 . . . . . . . . . 10  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  On )
9 onelon 4433 . . . . . . . . . . 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 4450 . . . . . . . . . . . . . 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 3403 . . . . . . . . . . . . . . . . 17  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  { x  e.  ~P A  |  x 
~<  A }
1615sseli 3189 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  { x  e.  ~P A  |  x  ~<  A }
)
17 breq1 4042 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
x  ~<  A  <->  z  ~<  A ) )
1817elrab 2936 . . . . . . . . . . . . . . . 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 3646 . . . . . . . . . . . . . 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 3202 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  C_  A )
25 vex 2804 . . . . . . . . . . . 12  |-  y  e. 
_V
2625elpw 3644 . . . . . . . . . . 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 2804 . . . . . . . . . . . 12  |-  z  e. 
_V
29 ssdomg 6923 . . . . . . . . . . . 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 7012 . . . . . . . . . . 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 4042 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
x  ~<  A  <->  y  ~<  A ) )
3635elrab 2936 . . . . . . . . . 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 3371 . . . . . . . . 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 1537 . . . . . . 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 4131 . . . . . . 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 4590 . . . . . 6  |-  Ord  On
44 trssord 4425 . . . . . 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 4416 . . . . 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 3203 . . . 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 6923 . . . . 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 4426 . . . . 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 4190 . . . . . . . . . 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 2432 . . . . . . . . 9  |-  F/_ x On
64 nfrab1 2733 . . . . . . . . 9  |-  F/_ x { x  e.  ~P A  |  x  ~<  A }
6563, 64nfin 3388 . . . . . . . 8  |-  F/_ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)
66 nfcv 2432 . . . . . . . 8  |-  F/_ x ~P A
67 nfcv 2432 . . . . . . . . 9  |-  F/_ x  ~<
68 nfcv 2432 . . . . . . . . 9  |-  F/_ x A
6965, 67, 68nfbr 4083 . . . . . . . 8  |-  F/ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  ~<  A
70 breq1 4042 . . . . . . . 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 2935 . . . . . . 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 3371 . . . . . 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 6908 . . 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 7595 . 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 1530    e. wcel 1696   {crab 2560   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039   Tr wtr 4129   Ord word 4407   Oncon0 4408   dom cdm 4705    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584
This theorem is referenced by:  tskwe2  8411  grothac  8468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588
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