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Theorem tskwe 8116
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 4473 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 rabexg 4439 . . . 4  |-  ( ~P A  e.  _V  ->  { x  e.  ~P A  |  x  ~<  A }  e.  _V )
3 incom 3540 . . . . 5  |-  ( { x  e.  ~P A  |  x  ~<  A }  i^i  On )  =  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )
4 inex1g 4432 . . . . 5  |-  ( { x  e.  ~P A  |  x  ~<  A }  e.  _V  ->  ( {
x  e.  ~P A  |  x  ~<  A }  i^i  On )  e.  _V )
53, 4syl5eqelr 2526 . . . 4  |-  ( { x  e.  ~P A  |  x  ~<  A }  e.  _V  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } )  e.  _V )
6 inss1 3567 . . . . . . . . . . 11  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  On
76sseli 3349 . . . . . . . . . 10  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  On )
8 onelon 4740 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
98ancoms 450 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  On )  ->  y  e.  On )
107, 9sylan2 471 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  On )
11 onelss 4757 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
1211impcom 430 . . . . . . . . . . . . 13  |-  ( ( y  e.  z  /\  z  e.  On )  ->  y  C_  z )
137, 12sylan2 471 . . . . . . . . . . . 12  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  C_  z )
14 inss2 3568 . . . . . . . . . . . . . . . . 17  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  { x  e.  ~P A  |  x 
~<  A }
1514sseli 3349 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  { x  e.  ~P A  |  x  ~<  A }
)
16 breq1 4292 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
x  ~<  A  <->  z  ~<  A ) )
1716elrab 3114 . . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . 14  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  ~P A )
2019elpwid 3867 . . . . . . . . . . . . 13  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  C_  A )
2120adantl 463 . . . . . . . . . . . 12  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
z  C_  A )
2213, 21sstrd 3363 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  C_  A )
23 selpw 3864 . . . . . . . . . . 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 2973 . . . . . . . . . . . 12  |-  z  e. 
_V
26 ssdomg 7351 . . . . . . . . . . . 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 460 . . . . . . . . . . . 12  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  ~<  A )
2928adantl 463 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
z  ~<  A )
30 domsdomtr 7442 . . . . . . . . . . 11  |-  ( ( y  ~<_  z  /\  z  ~<  A )  ->  y  ~<  A )
3127, 29, 30syl2anc 656 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  ~<  A )
32 breq1 4292 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
x  ~<  A  <->  y  ~<  A ) )
3332elrab 3114 . . . . . . . . . 10  |-  ( y  e.  { x  e. 
~P A  |  x 
~<  A }  <->  ( y  e.  ~P A  /\  y  ~<  A ) )
3424, 31, 33sylanbrc 659 . . . . . . . . 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 3537 . . . . . . . 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 1597 . . . . . . 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 4384 . . . . . . 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 6393 . . . . . 6  |-  Ord  On
40 trssord 4732 . . . . . 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 1309 . . . . 5  |-  Ord  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )
42 elong 4723 . . . . 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 462 . 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 458 . . . . 5  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  { x  e.  ~P A  |  x 
~<  A }  C_  A
)
4714, 46syl5ss 3364 . . . 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 7351 . . . . 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 462 . . . 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 4733 . . . . 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 1004 . . . . . 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 4452 . . . . . . . . . 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 462 . . . . . . . . 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 1003 . . . . . . 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 985 . . . . . . 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 2577 . . . . . . . . 9  |-  F/_ x On
60 nfrab1 2899 . . . . . . . . 9  |-  F/_ x { x  e.  ~P A  |  x  ~<  A }
6159, 60nfin 3554 . . . . . . . 8  |-  F/_ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)
62 nfcv 2577 . . . . . . . 8  |-  F/_ x ~P A
63 nfcv 2577 . . . . . . . . 9  |-  F/_ x  ~<
64 nfcv 2577 . . . . . . . . 9  |-  F/_ x A
6561, 63, 64nfbr 4333 . . . . . . . 8  |-  F/ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  ~<  A
66 breq1 4292 . . . . . . . 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 3112 . . . . . . 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 659 . . . . . 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 3537 . . . . 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 1184 . . . 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 7336 . . 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 659 . 2  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  ~~  A )
74 isnumi 8112 . 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 656 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 960   A.wal 1362    e. wcel 1761   {crab 2717   _Vcvv 2970    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   class class class wbr 4289   Tr wtr 4382   Ord word 4714   Oncon0 4715   dom cdm 4836    ~~ cen 7303    ~<_ cdom 7304    ~< csdm 7305   cardccrd 8101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-card 8105
This theorem is referenced by:  tskwe2  8936  grothac  8993
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