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Theorem tskwe 7826
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 4375 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 rabexg 4345 . . . . 5  |-  ( ~P A  e.  _V  ->  { x  e.  ~P A  |  x  ~<  A }  e.  _V )
3 incom 3525 . . . . . 6  |-  ( { x  e.  ~P A  |  x  ~<  A }  i^i  On )  =  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )
4 inex1g 4338 . . . . . 6  |-  ( { x  e.  ~P A  |  x  ~<  A }  e.  _V  ->  ( {
x  e.  ~P A  |  x  ~<  A }  i^i  On )  e.  _V )
53, 4syl5eqelr 2520 . . . . 5  |-  ( { x  e.  ~P A  |  x  ~<  A }  e.  _V  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } )  e.  _V )
61, 2, 53syl 19 . . . 4  |-  ( A  e.  V  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e. 
_V )
7 inss1 3553 . . . . . . . . . . 11  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  On
87sseli 3336 . . . . . . . . . 10  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  On )
9 onelon 4598 . . . . . . . . . . 11  |-  ( ( z  e.  On  /\  y  e.  z )  ->  y  e.  On )
109ancoms 440 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  On )  ->  y  e.  On )
118, 10sylan2 461 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  On )
12 onelss 4615 . . . . . . . . . . . . . 14  |-  ( z  e.  On  ->  (
y  e.  z  -> 
y  C_  z )
)
1312impcom 420 . . . . . . . . . . . . 13  |-  ( ( y  e.  z  /\  z  e.  On )  ->  y  C_  z )
148, 13sylan2 461 . . . . . . . . . . . 12  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  C_  z )
15 inss2 3554 . . . . . . . . . . . . . . . . 17  |-  ( On 
i^i  { x  e.  ~P A  |  x  ~<  A } )  C_  { x  e.  ~P A  |  x 
~<  A }
1615sseli 3336 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  { x  e.  ~P A  |  x  ~<  A }
)
17 breq1 4207 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
x  ~<  A  <->  z  ~<  A ) )
1817elrab 3084 . . . . . . . . . . . . . . . 16  |-  ( z  e.  { x  e. 
~P A  |  x 
~<  A }  <->  ( z  e.  ~P A  /\  z  ~<  A ) )
1916, 18sylib 189 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  ( z  e.  ~P A  /\  z  ~<  A ) )
2019simpld 446 . . . . . . . . . . . . . 14  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  e.  ~P A )
2120elpwid 3800 . . . . . . . . . . . . 13  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  C_  A )
2221adantl 453 . . . . . . . . . . . 12  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
z  C_  A )
2314, 22sstrd 3350 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  C_  A )
24 vex 2951 . . . . . . . . . . . 12  |-  y  e. 
_V
2524elpw 3797 . . . . . . . . . . 11  |-  ( y  e.  ~P A  <->  y  C_  A )
2623, 25sylibr 204 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  ~P A
)
27 vex 2951 . . . . . . . . . . . 12  |-  z  e. 
_V
28 ssdomg 7144 . . . . . . . . . . . 12  |-  ( z  e.  _V  ->  (
y  C_  z  ->  y  ~<_  z ) )
2927, 14, 28mpsyl 61 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  ~<_  z )
3019simprd 450 . . . . . . . . . . . 12  |-  ( z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  z  ~<  A )
3130adantl 453 . . . . . . . . . . 11  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
z  ~<  A )
32 domsdomtr 7233 . . . . . . . . . . 11  |-  ( ( y  ~<_  z  /\  z  ~<  A )  ->  y  ~<  A )
3329, 31, 32syl2anc 643 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  ~<  A )
34 breq1 4207 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
x  ~<  A  <->  y  ~<  A ) )
3534elrab 3084 . . . . . . . . . 10  |-  ( y  e.  { x  e. 
~P A  |  x 
~<  A }  <->  ( y  e.  ~P A  /\  y  ~<  A ) )
3626, 33, 35sylanbrc 646 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A } ) )  -> 
y  e.  { x  e.  ~P A  |  x 
~<  A } )
37 elin 3522 . . . . . . . . 9  |-  ( y  e.  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  <->  ( y  e.  On  /\  y  e. 
{ x  e.  ~P A  |  x  ~<  A } ) )
3811, 36, 37sylanbrc 646 . . . . . . . 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 } ) )
3938gen2 1556 . . . . . . 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 }
) )
40 dftr2 4296 . . . . . . 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 }
) ) )
4139, 40mpbir 201 . . . . . 6  |-  Tr  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )
42 ordon 4754 . . . . . 6  |-  Ord  On
43 trssord 4590 . . . . . 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 } ) )
4441, 7, 42, 43mp3an 1279 . . . . 5  |-  Ord  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )
45 elong 4581 . . . . 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 }
) ) )
4644, 45mpbiri 225 . . . 4  |-  ( ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  e. 
_V  ->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  e.  On )
476, 46syl 16 . . 3  |-  ( A  e.  V  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e.  On )
4847adantr 452 . 2  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e.  On )
49 simpr 448 . . . . 5  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  { x  e.  ~P A  |  x 
~<  A }  C_  A
)
5015, 49syl5ss 3351 . . . 4  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  C_  A )
51 ssdomg 7144 . . . . 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 ) )
5251adantr 452 . . . 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 ) )
5350, 52mpd 15 . . 3  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  ~<_  A )
54 ordirr 4591 . . . . 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 }
) )
5544, 54mp1i 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 }
) )
56473ad2ant1 978 . . . . . 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 )
57 elpw2g 4355 . . . . . . . . . 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 ) )
5857adantr 452 . . . . . . . . 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 ) )
5950, 58mpbird 224 . . . . . . . 8  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  e. 
~P A )
60593adant3 977 . . . . . . 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
)
61 simp3 959 . . . . . . 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 )
62 nfcv 2571 . . . . . . . . 9  |-  F/_ x On
63 nfrab1 2880 . . . . . . . . 9  |-  F/_ x { x  e.  ~P A  |  x  ~<  A }
6462, 63nfin 3539 . . . . . . . 8  |-  F/_ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)
65 nfcv 2571 . . . . . . . 8  |-  F/_ x ~P A
66 nfcv 2571 . . . . . . . . 9  |-  F/_ x  ~<
67 nfcv 2571 . . . . . . . . 9  |-  F/_ x A
6864, 66, 67nfbr 4248 . . . . . . . 8  |-  F/ x
( On  i^i  {
x  e.  ~P A  |  x  ~<  A }
)  ~<  A
69 breq1 4207 . . . . . . . 8  |-  ( x  =  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ->  ( x  ~<  A  <->  ( On  i^i  { x  e.  ~P A  |  x  ~<  A }
)  ~<  A ) )
7064, 65, 68, 69elrabf 3083 . . . . . . 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 ) )
7160, 61, 70sylanbrc 646 . . . . . 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 } )
72 elin 3522 . . . . . 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 } ) )
7356, 71, 72sylanbrc 646 . . . . 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 } ) )
74733expia 1155 . . . 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 }
) ) )
7555, 74mtod 170 . . 3  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  -.  ( On  i^i  { x  e.  ~P A  |  x 
~<  A } )  ~<  A )
76 bren2 7129 . . 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 ) )
7753, 75, 76sylanbrc 646 . 2  |-  ( ( A  e.  V  /\  { x  e.  ~P A  |  x  ~<  A }  C_  A )  ->  ( On  i^i  { x  e. 
~P A  |  x 
~<  A } )  ~~  A )
78 isnumi 7822 . 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 )
7948, 77, 78syl2anc 643 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 177    /\ wa 359    /\ w3a 936   A.wal 1549    e. wcel 1725   {crab 2701   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   class class class wbr 4204   Tr wtr 4294   Ord word 4572   Oncon0 4573   dom cdm 4869    ~~ cen 7097    ~<_ cdom 7098    ~< csdm 7099   cardccrd 7811
This theorem is referenced by:  tskwe2  8637  grothac  8694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-card 7815
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