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Theorem canthwe 8515
Description: The set of well-orders of a set  A strictly dominates  A. A stronger form of canth2 7251. 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 957 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  C_  A )
2 vex 2951 . . . . . . . . 9  |-  x  e. 
_V
32elpw 3797 . . . . . . . 8  |-  ( x  e.  ~P A  <->  x  C_  A
)
41, 3sylibr 204 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  ~P A )
5 simp2 958 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( x  X.  x
) )
6 xpss12 4972 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  x  C_  A )  -> 
( x  X.  x
)  C_  ( A  X.  A ) )
71, 1, 6syl2anc 643 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  X.  x ) 
C_  ( A  X.  A ) )
85, 7sstrd 3350 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( A  X.  A
) )
9 vex 2951 . . . . . . . . 9  |-  r  e. 
_V
109elpw 3797 . . . . . . . 8  |-  ( r  e.  ~P ( A  X.  A )  <->  r  C_  ( A  X.  A
) )
118, 10sylibr 204 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  e.  ~P ( A  X.  A ) )
124, 11jca 519 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) )
1312ssopab2i 4474 . . . . 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 4875 . . . . 5  |-  ( ~P A  X.  ~P ( A  X.  A ) )  =  { <. x ,  r >.  |  ( x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) }
1613, 14, 153sstr4i 3379 . . . 4  |-  O  C_  ( ~P A  X.  ~P ( A  X.  A
) )
17 pwexg 4375 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
18 xpexg 4980 . . . . . . 7  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
1918anidms 627 . . . . . 6  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
20 pwexg 4375 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  ~P ( A  X.  A
)  e.  _V )
2119, 20syl 16 . . . . 5  |-  ( A  e.  V  ->  ~P ( A  X.  A
)  e.  _V )
22 xpexg 4980 . . . . 5  |-  ( ( ~P A  e.  _V  /\ 
~P ( A  X.  A )  e.  _V )  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e. 
_V )
2317, 21, 22syl2anc 643 . . . 4  |-  ( A  e.  V  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e.  _V )
24 ssexg 4341 . . . 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 645 . . 3  |-  ( A  e.  V  ->  O  e.  _V )
26 simpr 448 . . . . . . . 8  |-  ( ( A  e.  V  /\  u  e.  A )  ->  u  e.  A )
2726snssd 3935 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  ->  { u }  C_  A )
28 0ss 3648 . . . . . . . 8  |-  (/)  C_  ( { u }  X.  { u } )
2928a1i 11 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  C_  ( { u }  X.  { u }
) )
30 rel0 4990 . . . . . . . 8  |-  Rel  (/)
31 noel 3624 . . . . . . . . . 10  |-  -.  <. u ,  u >.  e.  (/)
32 df-br 4205 . . . . . . . . . 10  |-  ( u
(/) u  <->  <. u ,  u >.  e.  (/) )
3331, 32mtbir 291 . . . . . . . . 9  |-  -.  u (/) u
34 wesn 4940 . . . . . . . . 9  |-  ( Rel  (/)  ->  ( (/)  We  {
u }  <->  -.  u (/) u ) )
3533, 34mpbiri 225 . . . . . . . 8  |-  ( Rel  (/)  ->  (/)  We  { u } )
3630, 35mp1i 12 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  We  { u } )
37 snex 4397 . . . . . . . 8  |-  { u }  e.  _V
38 0ex 4331 . . . . . . . 8  |-  (/)  e.  _V
39 simpl 444 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  x  =  {
u } )
4039sseq1d 3367 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  C_  A 
<->  { u }  C_  A ) )
41 simpr 448 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  r  =  (/) )
4239, 39xpeq12d 4894 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  X.  x )  =  ( { u }  X.  { u } ) )
4341, 42sseq12d 3369 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  C_  ( x  X.  x
)  <->  (/)  C_  ( {
u }  X.  {
u } ) ) )
44 weeq2 4563 . . . . . . . . . 10  |-  ( x  =  { u }  ->  ( r  We  x  <->  r  We  { u }
) )
45 weeq1 4562 . . . . . . . . . 10  |-  ( r  =  (/)  ->  ( r  We  { u }  <->  (/)  We 
{ u } ) )
4644, 45sylan9bb 681 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  We  x  <->  (/)  We  { u } ) )
4740, 43, 463anbi123d 1254 . . . . . . . 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 4463 . . . . . . 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 1138 . . . . . 6  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) } )
5049, 14syl6eleqr 2526 . . . . 5  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  O )
5150ex 424 . . . 4  |-  ( A  e.  V  ->  (
u  e.  A  ->  <. { u } ,  (/)
>.  e.  O ) )
52 eqid 2435 . . . . . . 7  |-  (/)  =  (/)
53 snex 4397 . . . . . . . 8  |-  { v }  e.  _V
5453, 38opth2 4430 . . . . . . 7  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  ( {
u }  =  {
v }  /\  (/)  =  (/) ) )
5552, 54mpbiran2 886 . . . . . 6  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  { u }  =  { v } )
56 vex 2951 . . . . . . 7  |-  u  e. 
_V
57 sneqbg 3961 . . . . . . 7  |-  ( u  e.  _V  ->  ( { u }  =  { v }  <->  u  =  v ) )
5856, 57ax-mp 8 . . . . . 6  |-  ( { u }  =  {
v }  <->  u  =  v )
5955, 58bitri 241 . . . . 5  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  u  =  v )
6059a1ii 25 . . . 4  |-  ( A  e.  V  ->  (
( u  e.  A  /\  v  e.  A
)  ->  ( <. { u } ,  (/) >.  =  <. { v } ,  (/) >.  <->  u  =  v
) ) )
6151, 60dom2d 7139 . . 3  |-  ( A  e.  V  ->  ( O  e.  _V  ->  A  ~<_  O ) )
6225, 61mpd 15 . 2  |-  ( A  e.  V  ->  A  ~<_  O )
63 eqid 2435 . . . . . . 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 8494 . . . . . 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 2435 . . . . . 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 2435 . . . . . 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 8514 . . . . 5  |-  ( A  e.  V  ->  -.  f : O -1-1-> A )
68 f1of1 5664 . . . . 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 1941 . . 3  |-  ( A  e.  V  ->  -.  E. f  f : O -1-1-onto-> A
)
71 ensym 7147 . . . 4  |-  ( A 
~~  O  ->  O  ~~  A )
72 bren 7108 . . . 4  |-  ( O 
~~  A  <->  E. f 
f : O -1-1-onto-> A )
7371, 72sylib 189 . . 3  |-  ( A 
~~  O  ->  E. f 
f : O -1-1-onto-> A )
7470, 73nsyl 115 . 2  |-  ( A  e.  V  ->  -.  A  ~~  O )
75 brsdom 7121 . 2  |-  ( A 
~<  O  <->  ( A  ~<_  O  /\  -.  A  ~~  O ) )
7662, 74, 75sylanbrc 646 1  |-  ( A  e.  V  ->  A  ~<  O )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   [.wsbc 3153    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   <.cop 3809   U.cuni 4007   class class class wbr 4204   {copab 4257    We wwe 4532    X. cxp 4867   `'ccnv 4868   dom cdm 4869   "cima 4872   Rel wrel 4874   -1-1->wf1 5442   -1-1-onto->wf1o 5444   ` cfv 5445  (class class class)co 6072    ~~ cen 7097    ~<_ cdom 7098    ~< csdm 7099
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-rep 4312  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-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-iun 4087  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-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  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-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-riota 6540  df-recs 6624  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-oi 7468
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