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Theorem canthp1lem1 8460
Description: Lemma for canthp1 8462. (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 7243 . . 3  |-  1o  ~<  2o
2 cdaxpdom 8002 . . 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 7175 . . . . . 6  |-  -.  1o  ~< 
(/)
5 breq2 4157 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
~<  A  <->  1o  ~<  (/) ) )
64, 5mtbiri 295 . . . . 5  |-  ( A  =  (/)  ->  -.  1o  ~<  A )
76con2i 114 . . . 4  |-  ( 1o 
~<  A  ->  -.  A  =  (/) )
8 neq0 3581 . . . 4  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
97, 8sylib 189 . . 3  |-  ( 1o 
~<  A  ->  E. x  x  e.  A )
10 relsdom 7052 . . . . . . . . . 10  |-  Rel  ~<
1110brrelex2i 4859 . . . . . . . . 9  |-  ( 1o 
~<  A  ->  A  e. 
_V )
1211adantr 452 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  e.  _V )
13 enrefg 7075 . . . . . . . 8  |-  ( A  e.  _V  ->  A  ~~  A )
1412, 13syl 16 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~~  A )
15 df2o2 6674 . . . . . . . . 9  |-  2o  =  { (/) ,  { (/) } }
16 pwpw0 3889 . . . . . . . . 9  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
1715, 16eqtr4i 2410 . . . . . . . 8  |-  2o  =  ~P { (/) }
18 0ex 4280 . . . . . . . . . 10  |-  (/)  e.  _V
19 vex 2902 . . . . . . . . . 10  |-  x  e. 
_V
20 en2sn 7122 . . . . . . . . . 10  |-  ( (
(/)  e.  _V  /\  x  e.  _V )  ->  { (/) } 
~~  { x }
)
2118, 19, 20mp2an 654 . . . . . . . . 9  |-  { (/) } 
~~  { x }
22 pwen 7216 . . . . . . . . 9  |-  ( {
(/) }  ~~  { x }  ->  ~P { (/) } 
~~  ~P { x }
)
2321, 22ax-mp 8 . . . . . . . 8  |-  ~P { (/)
}  ~~  ~P { x }
2417, 23eqbrtri 4172 . . . . . . 7  |-  2o  ~~  ~P { x }
25 xpen 7206 . . . . . . 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 4346 . . . . . . . 8  |-  { x }  e.  _V
2827pwex 4323 . . . . . . 7  |-  ~P {
x }  e.  _V
29 uncom 3434 . . . . . . . . 9  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( { x }  u.  ( A  \  {
x } ) )
30 simpr 448 . . . . . . . . . . 11  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  x  e.  A )
3130snssd 3886 . . . . . . . . . 10  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  { x }  C_  A )
32 undif 3651 . . . . . . . . . 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 2431 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  u.  { x }
)  =  A )
35 difexg 4292 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  \  { x }
)  e.  _V )
3612, 35syl 16 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  \  {
x } )  e. 
_V )
37 canth2g 7197 . . . . . . . . 9  |-  ( ( A  \  { x } )  e.  _V  ->  ( A  \  {
x } )  ~<  ~P ( A  \  {
x } ) )
38 domunsn 7193 . . . . . . . . 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 4175 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~<_  ~P ( A  \  { x } ) )
41 xpdom1g 7141 . . . . . . 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 7101 . . . . . 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 7998 . . . . . . 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 7094 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ~P ( A 
\  { x }
)  X.  ~P {
x } )  ~~  ~P ( ( A  \  { x } )  +c  { x }
) )
48 domentr 7102 . . . . 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 3476 . . . . . . . . 9  |-  ( ( A  \  { x } )  i^i  {
x } )  =  ( { x }  i^i  ( A  \  {
x } ) )
52 disjdif 3643 . . . . . . . . 9  |-  ( { x }  i^i  ( A  \  { x }
) )  =  (/)
5351, 52eqtri 2407 . . . . . . . 8  |-  ( ( A  \  { x } )  i^i  {
x } )  =  (/)
5453a1i 11 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  i^i  { x }
)  =  (/) )
55 cdaun 7985 . . . . . . 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 4177 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  +c  { x }
)  ~~  A )
58 pwen 7216 . . . . 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 7102 . . . 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 7096 . 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 2899    \ cdif 3260    u. cun 3261    i^i cin 3262    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   {csn 3757   {cpr 3758   class class class wbr 4153    X. cxp 4816  (class class class)co 6020   1oc1o 6653   2oc2o 6654    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044    +c ccda 7980
This theorem is referenced by:  canthp1lem2  8461  canthp1  8462
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-1o 6660  df-2o 6661  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-cda 7981
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