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Theorem canthp1lem1 8274
Description: Lemma for canthp1 8276. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
canthp1lem1  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ~P A )

Proof of Theorem canthp1lem1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 1sdom2 7061 . . 3  |-  1o  ~<  2o
2 cdaxpdom 7815 . . 3  |-  ( ( 1o  ~<  A  /\  1o  ~<  2o )  -> 
( A  +c  2o )  ~<_  ( A  X.  2o ) )
31, 2mpan2 652 . 2  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ( A  X.  2o ) )
4 sdom0 6993 . . . . . 6  |-  -.  1o  ~< 
(/)
5 breq2 4027 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
~<  A  <->  1o  ~<  (/) ) )
64, 5mtbiri 294 . . . . 5  |-  ( A  =  (/)  ->  -.  1o  ~<  A )
76con2i 112 . . . 4  |-  ( 1o 
~<  A  ->  -.  A  =  (/) )
8 neq0 3465 . . . 4  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
97, 8sylib 188 . . 3  |-  ( 1o 
~<  A  ->  E. x  x  e.  A )
10 relsdom 6870 . . . . . . . . . . . 12  |-  Rel  ~<
1110brrelex2i 4730 . . . . . . . . . . 11  |-  ( 1o 
~<  A  ->  A  e. 
_V )
1211adantr 451 . . . . . . . . . 10  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  e.  _V )
13 enrefg 6893 . . . . . . . . . 10  |-  ( A  e.  _V  ->  A  ~~  A )
1412, 13syl 15 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~~  A )
15 df2o2 6493 . . . . . . . . . . 11  |-  2o  =  { (/) ,  { (/) } }
16 pwpw0 3763 . . . . . . . . . . 11  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
1715, 16eqtr4i 2306 . . . . . . . . . 10  |-  2o  =  ~P { (/) }
18 0ex 4150 . . . . . . . . . . . 12  |-  (/)  e.  _V
19 vex 2791 . . . . . . . . . . . 12  |-  x  e. 
_V
20 en2sn 6940 . . . . . . . . . . . 12  |-  ( (
(/)  e.  _V  /\  x  e.  _V )  ->  { (/) } 
~~  { x }
)
2118, 19, 20mp2an 653 . . . . . . . . . . 11  |-  { (/) } 
~~  { x }
22 pwen 7034 . . . . . . . . . . 11  |-  ( {
(/) }  ~~  { x }  ->  ~P { (/) } 
~~  ~P { x }
)
2321, 22ax-mp 8 . . . . . . . . . 10  |-  ~P { (/)
}  ~~  ~P { x }
2417, 23eqbrtri 4042 . . . . . . . . 9  |-  2o  ~~  ~P { x }
25 xpen 7024 . . . . . . . . 9  |-  ( ( A  ~~  A  /\  2o  ~~  ~P { x } )  ->  ( A  X.  2o )  ~~  ( A  X.  ~P {
x } ) )
2614, 24, 25sylancl 643 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~~  ( A  X.  ~P { x } ) )
27 snex 4216 . . . . . . . . . 10  |-  { x }  e.  _V
2827pwex 4193 . . . . . . . . 9  |-  ~P {
x }  e.  _V
29 uncom 3319 . . . . . . . . . . 11  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( { x }  u.  ( A  \  {
x } ) )
30 simpr 447 . . . . . . . . . . . . 13  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  x  e.  A )
3130snssd 3760 . . . . . . . . . . . 12  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  { x }  C_  A )
32 undif 3534 . . . . . . . . . . . 12  |-  ( { x }  C_  A  <->  ( { x }  u.  ( A  \  { x } ) )  =  A )
3331, 32sylib 188 . . . . . . . . . . 11  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( { x }  u.  ( A  \  {
x } ) )  =  A )
3429, 33syl5eq 2327 . . . . . . . . . 10  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  u.  { x }
)  =  A )
35 difexg 4162 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  ( A  \  { x }
)  e.  _V )
3612, 35syl 15 . . . . . . . . . . 11  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  \  {
x } )  e. 
_V )
37 canth2g 7015 . . . . . . . . . . 11  |-  ( ( A  \  { x } )  e.  _V  ->  ( A  \  {
x } )  ~<  ~P ( A  \  {
x } ) )
38 domunsn 7011 . . . . . . . . . . 11  |-  ( ( A  \  { x } )  ~<  ~P ( A  \  { x }
)  ->  ( ( A  \  { x }
)  u.  { x } )  ~<_  ~P ( A  \  { x }
) )
3936, 37, 383syl 18 . . . . . . . . . 10  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  u.  { x }
)  ~<_  ~P ( A  \  { x } ) )
4034, 39eqbrtrrd 4045 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~<_  ~P ( A  \  { x } ) )
41 xpdom1g 6959 . . . . . . . . 9  |-  ( ( ~P { x }  e.  _V  /\  A  ~<_  ~P ( A  \  {
x } ) )  ->  ( A  X.  ~P { x } )  ~<_  ( ~P ( A 
\  { x }
)  X.  ~P {
x } ) )
4228, 40, 41sylancr 644 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  ~P { x } )  ~<_  ( ~P ( A 
\  { x }
)  X.  ~P {
x } ) )
43 endomtr 6919 . . . . . . . 8  |-  ( ( ( A  X.  2o )  ~~  ( A  X.  ~P { x } )  /\  ( A  X.  ~P { x } )  ~<_  ( ~P ( A 
\  { x }
)  X.  ~P {
x } ) )  ->  ( A  X.  2o )  ~<_  ( ~P ( A  \  { x } )  X.  ~P { x } ) )
4426, 42, 43syl2anc 642 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ( ~P ( A  \  { x }
)  X.  ~P {
x } ) )
45 pwcdaen 7811 . . . . . . . . 9  |-  ( ( ( A  \  {
x } )  e. 
_V  /\  { x }  e.  _V )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ( ~P ( A  \  {
x } )  X. 
~P { x }
) )
4636, 27, 45sylancl 643 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ( ~P ( A  \  {
x } )  X. 
~P { x }
) )
47 ensym 6910 . . . . . . . 8  |-  ( ~P ( ( A  \  { x } )  +c  { x }
)  ~~  ( ~P ( A  \  { x } )  X.  ~P { x } )  ->  ( ~P ( A  \  { x }
)  X.  ~P {
x } )  ~~  ~P ( ( A  \  { x } )  +c  { x }
) )
4846, 47syl 15 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ~P ( A 
\  { x }
)  X.  ~P {
x } )  ~~  ~P ( ( A  \  { x } )  +c  { x }
) )
49 domentr 6920 . . . . . . 7  |-  ( ( ( A  X.  2o )  ~<_  ( ~P ( A  \  { x }
)  X.  ~P {
x } )  /\  ( ~P ( A  \  { x } )  X.  ~P { x } )  ~~  ~P ( ( A  \  { x } )  +c  { x }
) )  ->  ( A  X.  2o )  ~<_  ~P ( ( A  \  { x } )  +c  { x }
) )
5044, 48, 49syl2anc 642 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ~P ( ( A 
\  { x }
)  +c  { x } ) )
5127a1i 10 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  { x }  e.  _V )
52 incom 3361 . . . . . . . . . . 11  |-  ( ( A  \  { x } )  i^i  {
x } )  =  ( { x }  i^i  ( A  \  {
x } ) )
53 disjdif 3526 . . . . . . . . . . 11  |-  ( { x }  i^i  ( A  \  { x }
) )  =  (/)
5452, 53eqtri 2303 . . . . . . . . . 10  |-  ( ( A  \  { x } )  i^i  {
x } )  =  (/)
5554a1i 10 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  i^i  { x }
)  =  (/) )
56 cdaun 7798 . . . . . . . . 9  |-  ( ( ( A  \  {
x } )  e. 
_V  /\  { x }  e.  _V  /\  (
( A  \  {
x } )  i^i 
{ x } )  =  (/) )  ->  (
( A  \  {
x } )  +c 
{ x } ) 
~~  ( ( A 
\  { x }
)  u.  { x } ) )
5736, 51, 55, 56syl3anc 1182 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  +c  { x }
)  ~~  ( ( A  \  { x }
)  u.  { x } ) )
5857, 34breqtrd 4047 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  +c  { x }
)  ~~  A )
59 pwen 7034 . . . . . . 7  |-  ( ( ( A  \  {
x } )  +c 
{ x } ) 
~~  A  ->  ~P ( ( A  \  { x } )  +c  { x }
)  ~~  ~P A
)
6058, 59syl 15 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ~P A )
61 domentr 6920 . . . . . 6  |-  ( ( ( A  X.  2o )  ~<_  ~P ( ( A 
\  { x }
)  +c  { x } )  /\  ~P ( ( A  \  { x } )  +c  { x }
)  ~~  ~P A
)  ->  ( A  X.  2o )  ~<_  ~P A
)
6250, 60, 61syl2anc 642 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ~P A )
6362ex 423 . . . 4  |-  ( 1o 
~<  A  ->  ( x  e.  A  ->  ( A  X.  2o )  ~<_  ~P A ) )
6463exlimdv 1664 . . 3  |-  ( 1o 
~<  A  ->  ( E. x  x  e.  A  ->  ( A  X.  2o )  ~<_  ~P A ) )
659, 64mpd 14 . 2  |-  ( 1o 
~<  A  ->  ( A  X.  2o )  ~<_  ~P A )
66 domtr 6914 . 2  |-  ( ( ( A  +c  2o )  ~<_  ( A  X.  2o )  /\  ( A  X.  2o )  ~<_  ~P A )  ->  ( A  +c  2o )  ~<_  ~P A )
673, 65, 66syl2anc 642 1  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   {cpr 3641   class class class wbr 4023    X. cxp 4687  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862    +c ccda 7793
This theorem is referenced by:  canthp1lem2  8275  canthp1  8276
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-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-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-int 3863  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-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-2o 6480  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-cda 7794
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