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Theorem canthwe 8273
Description: The set of well-orders of a set  A strictly dominates  A. A stronger form of canth2 7014. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)
Hypothesis
Ref Expression
canthwe.1  |-  O  =  { <. x ,  r
>.  |  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) }
Assertion
Ref Expression
canthwe  |-  ( A  e.  V  ->  A  ~<  O )
Distinct variable groups:    x, r, O    V, r, x    A, r, x

Proof of Theorem canthwe
Dummy variables  u  y  f  v  w  a  s  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 955 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  C_  A )
2 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
32elpw 3631 . . . . . . . 8  |-  ( x  e.  ~P A  <->  x  C_  A
)
41, 3sylibr 203 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  ~P A )
5 simp2 956 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( x  X.  x
) )
6 xpss12 4792 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  x  C_  A )  -> 
( x  X.  x
)  C_  ( A  X.  A ) )
71, 1, 6syl2anc 642 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  X.  x ) 
C_  ( A  X.  A ) )
85, 7sstrd 3189 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( A  X.  A
) )
9 vex 2791 . . . . . . . . 9  |-  r  e. 
_V
109elpw 3631 . . . . . . . 8  |-  ( r  e.  ~P ( A  X.  A )  <->  r  C_  ( A  X.  A
) )
118, 10sylibr 203 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  e.  ~P ( A  X.  A ) )
124, 11jca 518 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) )
1312ssopab2i 4292 . . . . 5  |-  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) }  C_  {
<. x ,  r >.  |  ( x  e. 
~P A  /\  r  e.  ~P ( A  X.  A ) ) }
14 canthwe.1 . . . . 5  |-  O  =  { <. x ,  r
>.  |  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) }
15 df-xp 4695 . . . . 5  |-  ( ~P A  X.  ~P ( A  X.  A ) )  =  { <. x ,  r >.  |  ( x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) }
1613, 14, 153sstr4i 3217 . . . 4  |-  O  C_  ( ~P A  X.  ~P ( A  X.  A
) )
17 pwexg 4194 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
18 xpexg 4800 . . . . . . 7  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
1918anidms 626 . . . . . 6  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
20 pwexg 4194 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  ~P ( A  X.  A
)  e.  _V )
2119, 20syl 15 . . . . 5  |-  ( A  e.  V  ->  ~P ( A  X.  A
)  e.  _V )
22 xpexg 4800 . . . . 5  |-  ( ( ~P A  e.  _V  /\ 
~P ( A  X.  A )  e.  _V )  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e. 
_V )
2317, 21, 22syl2anc 642 . . . 4  |-  ( A  e.  V  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e.  _V )
24 ssexg 4160 . . . 4  |-  ( ( O  C_  ( ~P A  X.  ~P ( A  X.  A ) )  /\  ( ~P A  X.  ~P ( A  X.  A ) )  e. 
_V )  ->  O  e.  _V )
2516, 23, 24sylancr 644 . . 3  |-  ( A  e.  V  ->  O  e.  _V )
26 simpr 447 . . . . . . . 8  |-  ( ( A  e.  V  /\  u  e.  A )  ->  u  e.  A )
2726snssd 3760 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  ->  { u }  C_  A )
28 0ss 3483 . . . . . . . 8  |-  (/)  C_  ( { u }  X.  { u } )
2928a1i 10 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  C_  ( { u }  X.  { u }
) )
30 rel0 4810 . . . . . . . 8  |-  Rel  (/)
31 noel 3459 . . . . . . . . . 10  |-  -.  <. u ,  u >.  e.  (/)
32 df-br 4024 . . . . . . . . . 10  |-  ( u
(/) u  <->  <. u ,  u >.  e.  (/) )
3331, 32mtbir 290 . . . . . . . . 9  |-  -.  u (/) u
34 wesn 4761 . . . . . . . . 9  |-  ( Rel  (/)  ->  ( (/)  We  {
u }  <->  -.  u (/) u ) )
3533, 34mpbiri 224 . . . . . . . 8  |-  ( Rel  (/)  ->  (/)  We  { u } )
3630, 35mp1i 11 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  We  { u } )
37 snex 4216 . . . . . . . 8  |-  { u }  e.  _V
38 0ex 4150 . . . . . . . 8  |-  (/)  e.  _V
39 simpl 443 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  x  =  {
u } )
4039sseq1d 3205 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  C_  A 
<->  { u }  C_  A ) )
41 simpr 447 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  r  =  (/) )
4239, 39xpeq12d 4714 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  X.  x )  =  ( { u }  X.  { u } ) )
4341, 42sseq12d 3207 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  C_  ( x  X.  x
)  <->  (/)  C_  ( {
u }  X.  {
u } ) ) )
44 weeq2 4382 . . . . . . . . . 10  |-  ( x  =  { u }  ->  ( r  We  x  <->  r  We  { u }
) )
45 weeq1 4381 . . . . . . . . . 10  |-  ( r  =  (/)  ->  ( r  We  { u }  <->  (/)  We 
{ u } ) )
4644, 45sylan9bb 680 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  We  x  <->  (/)  We  { u } ) )
4740, 43, 463anbi123d 1252 . . . . . . . 8  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x )  <->  ( {
u }  C_  A  /\  (/)  C_  ( {
u }  X.  {
u } )  /\  (/) 
We  { u }
) ) )
4837, 38, 47opelopaba 4281 . . . . . . 7  |-  ( <. { u } ,  (/)
>.  e.  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) }  <->  ( {
u }  C_  A  /\  (/)  C_  ( {
u }  X.  {
u } )  /\  (/) 
We  { u }
) )
4927, 29, 36, 48syl3anbrc 1136 . . . . . 6  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) } )
5049, 14syl6eleqr 2374 . . . . 5  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  O )
5150ex 423 . . . 4  |-  ( A  e.  V  ->  (
u  e.  A  ->  <. { u } ,  (/)
>.  e.  O ) )
52 eqid 2283 . . . . . . 7  |-  (/)  =  (/)
53 snex 4216 . . . . . . . 8  |-  { v }  e.  _V
5453, 38opth2 4248 . . . . . . 7  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  ( {
u }  =  {
v }  /\  (/)  =  (/) ) )
5552, 54mpbiran2 885 . . . . . 6  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  { u }  =  { v } )
56 vex 2791 . . . . . . 7  |-  u  e. 
_V
57 sneqbg 3783 . . . . . . 7  |-  ( u  e.  _V  ->  ( { u }  =  { v }  <->  u  =  v ) )
5856, 57ax-mp 8 . . . . . 6  |-  ( { u }  =  {
v }  <->  u  =  v )
5955, 58bitri 240 . . . . 5  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  u  =  v )
6059a1ii 24 . . . 4  |-  ( A  e.  V  ->  (
( u  e.  A  /\  v  e.  A
)  ->  ( <. { u } ,  (/) >.  =  <. { v } ,  (/) >.  <->  u  =  v
) ) )
6151, 60dom2d 6902 . . 3  |-  ( A  e.  V  ->  ( O  e.  _V  ->  A  ~<_  O ) )
6225, 61mpd 14 . 2  |-  ( A  e.  V  ->  A  ~<_  O )
63 eqid 2283 . . . . . . 7  |-  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }  =  { <. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v f ( s  i^i  (
v  X.  v ) ) )  =  z ) ) }
6463fpwwe2cbv 8252 . . . . . 6  |-  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }  =  { <. x ,  r >.  |  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  [. ( `' r " {
y } )  /  w ]. ( w f ( r  i^i  (
w  X.  w ) ) )  =  y ) ) }
65 eqid 2283 . . . . . 6  |-  U. dom  {
<. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v f ( s  i^i  (
v  X.  v ) ) )  =  z ) ) }  =  U. dom  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }
66 eqid 2283 . . . . . 6  |-  ( `' ( { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } `  U. dom  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } ) " { ( U. dom  {
<. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v f ( s  i^i  (
v  X.  v ) ) )  =  z ) ) } f ( { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } `  U. dom  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } ) ) } )  =  ( `' ( { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } `  U. dom  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } ) " { ( U. dom  {
<. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v f ( s  i^i  (
v  X.  v ) ) )  =  z ) ) } f ( { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } `  U. dom  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v f ( s  i^i  ( v  X.  v ) ) )  =  z ) ) } ) ) } )
6714, 64, 65, 66canthwelem 8272 . . . . 5  |-  ( A  e.  V  ->  -.  f : O -1-1-> A )
68 f1of1 5471 . . . . 5  |-  ( f : O -1-1-onto-> A  ->  f : O -1-1-> A )
6967, 68nsyl 113 . . . 4  |-  ( A  e.  V  ->  -.  f : O -1-1-onto-> A )
7069nexdv 1857 . . 3  |-  ( A  e.  V  ->  -.  E. f  f : O -1-1-onto-> A
)
71 ensym 6910 . . . 4  |-  ( A 
~~  O  ->  O  ~~  A )
72 bren 6871 . . . 4  |-  ( O 
~~  A  <->  E. f 
f : O -1-1-onto-> A )
7371, 72sylib 188 . . 3  |-  ( A 
~~  O  ->  E. f 
f : O -1-1-onto-> A )
7470, 73nsyl 113 . 2  |-  ( A  e.  V  ->  -.  A  ~~  O )
75 brsdom 6884 . 2  |-  ( A 
~<  O  <->  ( A  ~<_  O  /\  -.  A  ~~  O ) )
7662, 74, 75sylanbrc 645 1  |-  ( A  e.  V  ->  A  ~<  O )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   <.cop 3643   U.cuni 3827   class class class wbr 4023   {copab 4076    We wwe 4351    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692   Rel wrel 4694   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
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-rep 4131  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-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  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-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-riota 6304  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225
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