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Theorem canthwe 8268
Description: The set of well-orders of a set  A strictly dominates  A. A stronger form of canth2 7009. 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
StepHypRef Expression
1 simp1 960 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  C_  A )
2 vex 2792 . . . . . . . . 9  |-  x  e. 
_V
32elpw 3632 . . . . . . . 8  |-  ( x  e.  ~P A  <->  x  C_  A
)
41, 3sylibr 205 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  ~P A )
5 simp2 961 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( x  X.  x
) )
6 xpss12 4791 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  x  C_  A )  -> 
( x  X.  x
)  C_  ( A  X.  A ) )
71, 1, 6syl2anc 645 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  X.  x ) 
C_  ( A  X.  A ) )
85, 7sstrd 3190 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( A  X.  A
) )
9 vex 2792 . . . . . . . . 9  |-  r  e. 
_V
109elpw 3632 . . . . . . . 8  |-  ( r  e.  ~P ( A  X.  A )  <->  r  C_  ( A  X.  A
) )
118, 10sylibr 205 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  e.  ~P ( A  X.  A ) )
124, 11jca 520 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) )
1312ssopab2i 4291 . . . . 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 4694 . . . . 5  |-  ( ~P A  X.  ~P ( A  X.  A ) )  =  { <. x ,  r >.  |  ( x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) }
1613, 14, 153sstr4i 3218 . . . 4  |-  O  C_  ( ~P A  X.  ~P ( A  X.  A
) )
17 pwexg 4193 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
18 xpexg 4799 . . . . . . 7  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
1918anidms 629 . . . . . 6  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
20 pwexg 4193 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  ~P ( A  X.  A
)  e.  _V )
2119, 20syl 17 . . . . 5  |-  ( A  e.  V  ->  ~P ( A  X.  A
)  e.  _V )
22 xpexg 4799 . . . . 5  |-  ( ( ~P A  e.  _V  /\ 
~P ( A  X.  A )  e.  _V )  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e. 
_V )
2317, 21, 22syl2anc 645 . . . 4  |-  ( A  e.  V  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e.  _V )
24 ssexg 4161 . . . 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 647 . . 3  |-  ( A  e.  V  ->  O  e.  _V )
26 simpr 449 . . . . . . . 8  |-  ( ( A  e.  V  /\  u  e.  A )  ->  u  e.  A )
2726snssd 3761 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  ->  { u }  C_  A )
28 0ss 3484 . . . . . . . 8  |-  (/)  C_  ( { u }  X.  { u } )
2928a1i 12 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  C_  ( { u }  X.  { u }
) )
30 rel0 4809 . . . . . . . 8  |-  Rel  (/)
31 noel 3460 . . . . . . . . . 10  |-  -.  <. u ,  u >.  e.  (/)
32 df-br 4025 . . . . . . . . . 10  |-  ( u
(/) u  <->  <. u ,  u >.  e.  (/) )
3331, 32mtbir 292 . . . . . . . . 9  |-  -.  u (/) u
34 wesn 4760 . . . . . . . . 9  |-  ( Rel  (/)  ->  ( (/)  We  {
u }  <->  -.  u (/) u ) )
3533, 34mpbiri 226 . . . . . . . 8  |-  ( Rel  (/)  ->  (/)  We  { u } )
3630, 35mp1i 13 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  We  { u } )
37 snex 4215 . . . . . . . 8  |-  { u }  e.  _V
38 0ex 4151 . . . . . . . 8  |-  (/)  e.  _V
39 simpl 445 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  x  =  {
u } )
4039sseq1d 3206 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  C_  A 
<->  { u }  C_  A ) )
41 simpr 449 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  r  =  (/) )
4239, 39xpeq12d 4713 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  X.  x )  =  ( { u }  X.  { u } ) )
4341, 42sseq12d 3208 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  C_  ( x  X.  x
)  <->  (/)  C_  ( {
u }  X.  {
u } ) ) )
44 weeq2 4381 . . . . . . . . . 10  |-  ( x  =  { u }  ->  ( r  We  x  <->  r  We  { u }
) )
45 weeq1 4380 . . . . . . . . . 10  |-  ( r  =  (/)  ->  ( r  We  { u }  <->  (/)  We 
{ u } ) )
4644, 45sylan9bb 683 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  We  x  <->  (/)  We  { u } ) )
4740, 43, 463anbi123d 1257 . . . . . . . 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 4280 . . . . . . 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 1141 . . . . . 6  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) } )
5049, 14syl6eleqr 2375 . . . . 5  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  O )
5150ex 425 . . . 4  |-  ( A  e.  V  ->  (
u  e.  A  ->  <. { u } ,  (/)
>.  e.  O ) )
52 eqid 2284 . . . . . . 7  |-  (/)  =  (/)
53 snex 4215 . . . . . . . 8  |-  { v }  e.  _V
5453, 38opth2 4247 . . . . . . 7  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  ( {
u }  =  {
v }  /\  (/)  =  (/) ) )
5552, 54mpbiran2 890 . . . . . 6  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  { u }  =  { v } )
56 vex 2792 . . . . . . 7  |-  u  e. 
_V
57 sneqbg 3784 . . . . . . 7  |-  ( u  e.  _V  ->  ( { u }  =  { v }  <->  u  =  v ) )
5856, 57ax-mp 10 . . . . . 6  |-  ( { u }  =  {
v }  <->  u  =  v )
5955, 58bitri 242 . . . . 5  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  u  =  v )
6059a1ii 26 . . . 4  |-  ( A  e.  V  ->  (
( u  e.  A  /\  v  e.  A
)  ->  ( <. { u } ,  (/) >.  =  <. { v } ,  (/) >.  <->  u  =  v
) ) )
6151, 60dom2d 6897 . . 3  |-  ( A  e.  V  ->  ( O  e.  _V  ->  A  ~<_  O ) )
6225, 61mpd 16 . 2  |-  ( A  e.  V  ->  A  ~<_  O )
63 eqid 2284 . . . . . . 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 8247 . . . . . 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 2284 . . . . . 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 2284 . . . . . 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 8267 . . . . 5  |-  ( A  e.  V  ->  -.  f : O -1-1-> A )
68 f1of1 5436 . . . . 5  |-  ( f : O -1-1-onto-> A  ->  f : O -1-1-> A )
6967, 68nsyl 115 . . . 4  |-  ( A  e.  V  ->  -.  f : O -1-1-onto-> A )
7069nexdv 1868 . . 3  |-  ( A  e.  V  ->  -.  E. f  f : O -1-1-onto-> A
)
71 ensym 6905 . . . 4  |-  ( A 
~~  O  ->  O  ~~  A )
72 bren 6866 . . . 4  |-  ( O 
~~  A  <->  E. f 
f : O -1-1-onto-> A )
7371, 72sylib 190 . . 3  |-  ( A 
~~  O  ->  E. f 
f : O -1-1-onto-> A )
7470, 73nsyl 115 . 2  |-  ( A  e.  V  ->  -.  A  ~~  O )
75 brsdom 6879 . 2  |-  ( A 
~<  O  <->  ( A  ~<_  O  /\  -.  A  ~~  O ) )
7662, 74, 75sylanbrc 648 1  |-  ( A  e.  V  ->  A  ~<  O )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939   E.wex 1533    = wceq 1628    e. wcel 1688   A.wral 2544   _Vcvv 2789   [.wsbc 2992    i^i cin 3152    C_ wss 3153   (/)c0 3456   ~Pcpw 3626   {csn 3641   <.cop 3644   U.cuni 3828   class class class wbr 4024   {copab 4077    We wwe 4350    X. cxp 4686   `'ccnv 4687   dom cdm 4688   "cima 4691   Rel wrel 4693   -1-1->wf1 5218   -1-1-onto->wf1o 5220   ` cfv 5221  (class class class)co 5819    ~~ cen 6855    ~<_ cdom 6856    ~< csdm 6857
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-iota 6252  df-riota 6299  df-recs 6383  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-oi 7220
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