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Theorem canthwe 9046
Description: The set of well-orders of a set  A strictly dominates  A. A stronger form of canth2 7689. 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 996 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  C_  A )
2 selpw 4022 . . . . . . . 8  |-  ( x  e.  ~P A  <->  x  C_  A
)
31, 2sylibr 212 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  ~P A )
4 simp2 997 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( x  X.  x
) )
5 xpss12 5117 . . . . . . . . . 10  |-  ( ( x  C_  A  /\  x  C_  A )  -> 
( x  X.  x
)  C_  ( A  X.  A ) )
61, 1, 5syl2anc 661 . . . . . . . . 9  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  X.  x ) 
C_  ( A  X.  A ) )
74, 6sstrd 3509 . . . . . . . 8  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  C_  ( A  X.  A
) )
8 selpw 4022 . . . . . . . 8  |-  ( r  e.  ~P ( A  X.  A )  <->  r  C_  ( A  X.  A
) )
97, 8sylibr 212 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  r  e.  ~P ( A  X.  A ) )
103, 9jca 532 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  (
x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) )
1110ssopab2i 4784 . . . . 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 ) ) }
12 canthwe.1 . . . . 5  |-  O  =  { <. x ,  r
>.  |  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) }
13 df-xp 5014 . . . . 5  |-  ( ~P A  X.  ~P ( A  X.  A ) )  =  { <. x ,  r >.  |  ( x  e.  ~P A  /\  r  e.  ~P ( A  X.  A
) ) }
1411, 12, 133sstr4i 3538 . . . 4  |-  O  C_  ( ~P A  X.  ~P ( A  X.  A
) )
15 pwexg 4640 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
16 sqxpexg 6604 . . . . . 6  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
17 pwexg 4640 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  ~P ( A  X.  A
)  e.  _V )
1816, 17syl 16 . . . . 5  |-  ( A  e.  V  ->  ~P ( A  X.  A
)  e.  _V )
19 xpexg 6601 . . . . 5  |-  ( ( ~P A  e.  _V  /\ 
~P ( A  X.  A )  e.  _V )  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e. 
_V )
2015, 18, 19syl2anc 661 . . . 4  |-  ( A  e.  V  ->  ( ~P A  X.  ~P ( A  X.  A ) )  e.  _V )
21 ssexg 4602 . . . 4  |-  ( ( O  C_  ( ~P A  X.  ~P ( A  X.  A ) )  /\  ( ~P A  X.  ~P ( A  X.  A ) )  e. 
_V )  ->  O  e.  _V )
2214, 20, 21sylancr 663 . . 3  |-  ( A  e.  V  ->  O  e.  _V )
23 simpr 461 . . . . . . . 8  |-  ( ( A  e.  V  /\  u  e.  A )  ->  u  e.  A )
2423snssd 4177 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  ->  { u }  C_  A )
25 0ss 3823 . . . . . . . 8  |-  (/)  C_  ( { u }  X.  { u } )
2625a1i 11 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  C_  ( { u }  X.  { u }
) )
27 rel0 5136 . . . . . . . 8  |-  Rel  (/)
28 br0 4502 . . . . . . . . 9  |-  -.  u (/) u
29 wesn 5080 . . . . . . . . 9  |-  ( Rel  (/)  ->  ( (/)  We  {
u }  <->  -.  u (/) u ) )
3028, 29mpbiri 233 . . . . . . . 8  |-  ( Rel  (/)  ->  (/)  We  { u } )
3127, 30mp1i 12 . . . . . . 7  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
(/)  We  { u } )
32 snex 4697 . . . . . . . 8  |-  { u }  e.  _V
33 0ex 4587 . . . . . . . 8  |-  (/)  e.  _V
34 simpl 457 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  x  =  {
u } )
3534sseq1d 3526 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  C_  A 
<->  { u }  C_  A ) )
36 simpr 461 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  r  =  (/) )
3734sqxpeqd 5034 . . . . . . . . . 10  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( x  X.  x )  =  ( { u }  X.  { u } ) )
3836, 37sseq12d 3528 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  C_  ( x  X.  x
)  <->  (/)  C_  ( {
u }  X.  {
u } ) ) )
39 weeq2 4877 . . . . . . . . . 10  |-  ( x  =  { u }  ->  ( r  We  x  <->  r  We  { u }
) )
40 weeq1 4876 . . . . . . . . . 10  |-  ( r  =  (/)  ->  ( r  We  { u }  <->  (/)  We 
{ u } ) )
4139, 40sylan9bb 699 . . . . . . . . 9  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( r  We  x  <->  (/)  We  { u } ) )
4235, 38, 413anbi123d 1299 . . . . . . . 8  |-  ( ( x  =  { u }  /\  r  =  (/) )  ->  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x )  <->  ( {
u }  C_  A  /\  (/)  C_  ( {
u }  X.  {
u } )  /\  (/) 
We  { u }
) ) )
4332, 33, 42opelopaba 4772 . . . . . . 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 }
) )
4424, 26, 31, 43syl3anbrc 1180 . . . . . 6  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) } )
4544, 12syl6eleqr 2556 . . . . 5  |-  ( ( A  e.  V  /\  u  e.  A )  -> 
<. { u } ,  (/)
>.  e.  O )
4645ex 434 . . . 4  |-  ( A  e.  V  ->  (
u  e.  A  ->  <. { u } ,  (/)
>.  e.  O ) )
47 eqid 2457 . . . . . . 7  |-  (/)  =  (/)
48 snex 4697 . . . . . . . 8  |-  { v }  e.  _V
4948, 33opth2 4734 . . . . . . 7  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  ( {
u }  =  {
v }  /\  (/)  =  (/) ) )
5047, 49mpbiran2 919 . . . . . 6  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  { u }  =  { v } )
51 vex 3112 . . . . . . 7  |-  u  e. 
_V
52 sneqbg 4202 . . . . . . 7  |-  ( u  e.  _V  ->  ( { u }  =  { v }  <->  u  =  v ) )
5351, 52ax-mp 5 . . . . . 6  |-  ( { u }  =  {
v }  <->  u  =  v )
5450, 53bitri 249 . . . . 5  |-  ( <. { u } ,  (/)
>.  =  <. { v } ,  (/) >.  <->  u  =  v )
5554a1ii 27 . . . 4  |-  ( A  e.  V  ->  (
( u  e.  A  /\  v  e.  A
)  ->  ( <. { u } ,  (/) >.  =  <. { v } ,  (/) >.  <->  u  =  v
) ) )
5646, 55dom2d 7575 . . 3  |-  ( A  e.  V  ->  ( O  e.  _V  ->  A  ~<_  O ) )
5722, 56mpd 15 . 2  |-  ( A  e.  V  ->  A  ~<_  O )
58 eqid 2457 . . . . . . 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 ) ) }
5958fpwwe2cbv 9025 . . . . . 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 ) ) }
60 eqid 2457 . . . . . 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 ) ) }
61 eqid 2457 . . . . . 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 ) ) } ) ) } )
6212, 59, 60, 61canthwelem 9045 . . . . 5  |-  ( A  e.  V  ->  -.  f : O -1-1-> A )
63 f1of1 5821 . . . . 5  |-  ( f : O -1-1-onto-> A  ->  f : O -1-1-> A )
6462, 63nsyl 121 . . . 4  |-  ( A  e.  V  ->  -.  f : O -1-1-onto-> A )
6564nexdv 1885 . . 3  |-  ( A  e.  V  ->  -.  E. f  f : O -1-1-onto-> A
)
66 ensym 7583 . . . 4  |-  ( A 
~~  O  ->  O  ~~  A )
67 bren 7544 . . . 4  |-  ( O 
~~  A  <->  E. f 
f : O -1-1-onto-> A )
6866, 67sylib 196 . . 3  |-  ( A 
~~  O  ->  E. f 
f : O -1-1-onto-> A )
6965, 68nsyl 121 . 2  |-  ( A  e.  V  ->  -.  A  ~~  O )
70 brsdom 7557 . 2  |-  ( A 
~<  O  <->  ( A  ~<_  O  /\  -.  A  ~~  O ) )
7157, 69, 70sylanbrc 664 1  |-  ( A  e.  V  ->  A  ~<  O )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   A.wral 2807   _Vcvv 3109   [.wsbc 3327    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   <.cop 4038   U.cuni 4251   class class class wbr 4456   {copab 4514    We wwe 4846    X. cxp 5006   `'ccnv 5007   dom cdm 5008   "cima 5011   Rel wrel 5013   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    ~~ cen 7532    ~<_ cdom 7533    ~< csdm 7534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-recs 7060  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-oi 7953
This theorem is referenced by: (None)
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