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Theorem canthp1lem1 8453
Description: Lemma for canthp1 8455. (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 7236 . . 3  |-  1o  ~<  2o
2 cdaxpdom 7995 . . 3  |-  ( ( 1o  ~<  A  /\  1o  ~<  2o )  -> 
( A  +c  2o )  ~<_  ( A  X.  2o ) )
31, 2mpan2 653 . 2  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ( A  X.  2o ) )
4 sdom0 7168 . . . . . 6  |-  -.  1o  ~< 
(/)
5 breq2 4150 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
~<  A  <->  1o  ~<  (/) ) )
64, 5mtbiri 295 . . . . 5  |-  ( A  =  (/)  ->  -.  1o  ~<  A )
76con2i 114 . . . 4  |-  ( 1o 
~<  A  ->  -.  A  =  (/) )
8 neq0 3574 . . . 4  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
97, 8sylib 189 . . 3  |-  ( 1o 
~<  A  ->  E. x  x  e.  A )
10 relsdom 7045 . . . . . . . . . 10  |-  Rel  ~<
1110brrelex2i 4852 . . . . . . . . 9  |-  ( 1o 
~<  A  ->  A  e. 
_V )
1211adantr 452 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  e.  _V )
13 enrefg 7068 . . . . . . . 8  |-  ( A  e.  _V  ->  A  ~~  A )
1412, 13syl 16 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~~  A )
15 df2o2 6667 . . . . . . . . 9  |-  2o  =  { (/) ,  { (/) } }
16 pwpw0 3882 . . . . . . . . 9  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
1715, 16eqtr4i 2403 . . . . . . . 8  |-  2o  =  ~P { (/) }
18 0ex 4273 . . . . . . . . . 10  |-  (/)  e.  _V
19 vex 2895 . . . . . . . . . 10  |-  x  e. 
_V
20 en2sn 7115 . . . . . . . . . 10  |-  ( (
(/)  e.  _V  /\  x  e.  _V )  ->  { (/) } 
~~  { x }
)
2118, 19, 20mp2an 654 . . . . . . . . 9  |-  { (/) } 
~~  { x }
22 pwen 7209 . . . . . . . . 9  |-  ( {
(/) }  ~~  { x }  ->  ~P { (/) } 
~~  ~P { x }
)
2321, 22ax-mp 8 . . . . . . . 8  |-  ~P { (/)
}  ~~  ~P { x }
2417, 23eqbrtri 4165 . . . . . . 7  |-  2o  ~~  ~P { x }
25 xpen 7199 . . . . . . 7  |-  ( ( A  ~~  A  /\  2o  ~~  ~P { x } )  ->  ( A  X.  2o )  ~~  ( A  X.  ~P {
x } ) )
2614, 24, 25sylancl 644 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~~  ( A  X.  ~P { x } ) )
27 snex 4339 . . . . . . . 8  |-  { x }  e.  _V
2827pwex 4316 . . . . . . 7  |-  ~P {
x }  e.  _V
29 uncom 3427 . . . . . . . . 9  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( { x }  u.  ( A  \  {
x } ) )
30 simpr 448 . . . . . . . . . . 11  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  x  e.  A )
3130snssd 3879 . . . . . . . . . 10  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  { x }  C_  A )
32 undif 3644 . . . . . . . . . 10  |-  ( { x }  C_  A  <->  ( { x }  u.  ( A  \  { x } ) )  =  A )
3331, 32sylib 189 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( { x }  u.  ( A  \  {
x } ) )  =  A )
3429, 33syl5eq 2424 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  u.  { x }
)  =  A )
35 difexg 4285 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  \  { x }
)  e.  _V )
3612, 35syl 16 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  \  {
x } )  e. 
_V )
37 canth2g 7190 . . . . . . . . 9  |-  ( ( A  \  { x } )  e.  _V  ->  ( A  \  {
x } )  ~<  ~P ( A  \  {
x } ) )
38 domunsn 7186 . . . . . . . . 9  |-  ( ( A  \  { x } )  ~<  ~P ( A  \  { x }
)  ->  ( ( A  \  { x }
)  u.  { x } )  ~<_  ~P ( A  \  { x }
) )
3936, 37, 383syl 19 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  u.  { x }
)  ~<_  ~P ( A  \  { x } ) )
4034, 39eqbrtrrd 4168 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~<_  ~P ( A  \  { x } ) )
41 xpdom1g 7134 . . . . . . 7  |-  ( ( ~P { x }  e.  _V  /\  A  ~<_  ~P ( A  \  {
x } ) )  ->  ( A  X.  ~P { x } )  ~<_  ( ~P ( A 
\  { x }
)  X.  ~P {
x } ) )
4228, 40, 41sylancr 645 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  ~P { x } )  ~<_  ( ~P ( A 
\  { x }
)  X.  ~P {
x } ) )
43 endomtr 7094 . . . . . 6  |-  ( ( ( 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 643 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ( ~P ( A  \  { x }
)  X.  ~P {
x } ) )
45 pwcdaen 7991 . . . . . . 7  |-  ( ( ( A  \  {
x } )  e. 
_V  /\  { x }  e.  _V )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ( ~P ( A  \  {
x } )  X. 
~P { x }
) )
4636, 27, 45sylancl 644 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ( ~P ( A  \  {
x } )  X. 
~P { x }
) )
4746ensymd 7087 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ~P ( A 
\  { x }
)  X.  ~P {
x } )  ~~  ~P ( ( A  \  { x } )  +c  { x }
) )
48 domentr 7095 . . . . 5  |-  ( ( ( 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 }
) )
4944, 47, 48syl2anc 643 . . . 4  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ~P ( ( A 
\  { x }
)  +c  { x } ) )
5027a1i 11 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  { x }  e.  _V )
51 incom 3469 . . . . . . . . 9  |-  ( ( A  \  { x } )  i^i  {
x } )  =  ( { x }  i^i  ( A  \  {
x } ) )
52 disjdif 3636 . . . . . . . . 9  |-  ( { x }  i^i  ( A  \  { x }
) )  =  (/)
5351, 52eqtri 2400 . . . . . . . 8  |-  ( ( A  \  { x } )  i^i  {
x } )  =  (/)
5453a1i 11 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  i^i  { x }
)  =  (/) )
55 cdaun 7978 . . . . . . 7  |-  ( ( ( A  \  {
x } )  e. 
_V  /\  { x }  e.  _V  /\  (
( A  \  {
x } )  i^i 
{ x } )  =  (/) )  ->  (
( A  \  {
x } )  +c 
{ x } ) 
~~  ( ( A 
\  { x }
)  u.  { x } ) )
5636, 50, 54, 55syl3anc 1184 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  +c  { x }
)  ~~  ( ( A  \  { x }
)  u.  { x } ) )
5756, 34breqtrd 4170 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  +c  { x }
)  ~~  A )
58 pwen 7209 . . . . 5  |-  ( ( ( A  \  {
x } )  +c 
{ x } ) 
~~  A  ->  ~P ( ( A  \  { x } )  +c  { x }
)  ~~  ~P A
)
5957, 58syl 16 . . . 4  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ~P A )
60 domentr 7095 . . . 4  |-  ( ( ( A  X.  2o )  ~<_  ~P ( ( A 
\  { x }
)  +c  { x } )  /\  ~P ( ( A  \  { x } )  +c  { x }
)  ~~  ~P A
)  ->  ( A  X.  2o )  ~<_  ~P A
)
6149, 59, 60syl2anc 643 . . 3  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ~P A )
629, 61exlimddv 1645 . 2  |-  ( 1o 
~<  A  ->  ( A  X.  2o )  ~<_  ~P A )
63 domtr 7089 . 2  |-  ( ( ( A  +c  2o )  ~<_  ( A  X.  2o )  /\  ( A  X.  2o )  ~<_  ~P A )  ->  ( A  +c  2o )  ~<_  ~P A )
643, 62, 63syl2anc 643 1  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2892    \ cdif 3253    u. cun 3254    i^i cin 3255    C_ wss 3256   (/)c0 3564   ~Pcpw 3735   {csn 3750   {cpr 3751   class class class wbr 4146    X. cxp 4809  (class class class)co 6013   1oc1o 6646   2oc2o 6647    ~~ cen 7035    ~<_ cdom 7036    ~< csdm 7037    +c ccda 7973
This theorem is referenced by:  canthp1lem2  8454  canthp1  8455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-1o 6653  df-2o 6654  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-cda 7974
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