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Theorem canthp1lem1 9019
Description: Lemma for canthp1 9021. (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 7711 . . 3  |-  1o  ~<  2o
2 cdaxpdom 8560 . . 3  |-  ( ( 1o  ~<  A  /\  1o  ~<  2o )  -> 
( A  +c  2o )  ~<_  ( A  X.  2o ) )
31, 2mpan2 669 . 2  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ( A  X.  2o ) )
4 sdom0 7642 . . . . . 6  |-  -.  1o  ~< 
(/)
5 breq2 4443 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
~<  A  <->  1o  ~<  (/) ) )
64, 5mtbiri 301 . . . . 5  |-  ( A  =  (/)  ->  -.  1o  ~<  A )
76con2i 120 . . . 4  |-  ( 1o 
~<  A  ->  -.  A  =  (/) )
8 neq0 3794 . . . 4  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
97, 8sylib 196 . . 3  |-  ( 1o 
~<  A  ->  E. x  x  e.  A )
10 relsdom 7516 . . . . . . . . . 10  |-  Rel  ~<
1110brrelex2i 5030 . . . . . . . . 9  |-  ( 1o 
~<  A  ->  A  e. 
_V )
1211adantr 463 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  e.  _V )
13 enrefg 7540 . . . . . . . 8  |-  ( A  e.  _V  ->  A  ~~  A )
1412, 13syl 16 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~~  A )
15 df2o2 7136 . . . . . . . . 9  |-  2o  =  { (/) ,  { (/) } }
16 pwpw0 4164 . . . . . . . . 9  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
1715, 16eqtr4i 2486 . . . . . . . 8  |-  2o  =  ~P { (/) }
18 0ex 4569 . . . . . . . . . 10  |-  (/)  e.  _V
19 vex 3109 . . . . . . . . . 10  |-  x  e. 
_V
20 en2sn 7588 . . . . . . . . . 10  |-  ( (
(/)  e.  _V  /\  x  e.  _V )  ->  { (/) } 
~~  { x }
)
2118, 19, 20mp2an 670 . . . . . . . . 9  |-  { (/) } 
~~  { x }
22 pwen 7683 . . . . . . . . 9  |-  ( {
(/) }  ~~  { x }  ->  ~P { (/) } 
~~  ~P { x }
)
2321, 22ax-mp 5 . . . . . . . 8  |-  ~P { (/)
}  ~~  ~P { x }
2417, 23eqbrtri 4458 . . . . . . 7  |-  2o  ~~  ~P { x }
25 xpen 7673 . . . . . . 7  |-  ( ( A  ~~  A  /\  2o  ~~  ~P { x } )  ->  ( A  X.  2o )  ~~  ( A  X.  ~P {
x } ) )
2614, 24, 25sylancl 660 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~~  ( A  X.  ~P { x } ) )
27 snex 4678 . . . . . . . 8  |-  { x }  e.  _V
2827pwex 4620 . . . . . . 7  |-  ~P {
x }  e.  _V
29 uncom 3634 . . . . . . . . 9  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( { x }  u.  ( A  \  {
x } ) )
30 simpr 459 . . . . . . . . . . 11  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  x  e.  A )
3130snssd 4161 . . . . . . . . . 10  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  { x }  C_  A )
32 undif 3896 . . . . . . . . . 10  |-  ( { x }  C_  A  <->  ( { x }  u.  ( A  \  { x } ) )  =  A )
3331, 32sylib 196 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( { x }  u.  ( A  \  {
x } ) )  =  A )
3429, 33syl5eq 2507 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  u.  { x }
)  =  A )
35 difexg 4585 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  \  { x }
)  e.  _V )
3612, 35syl 16 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  \  {
x } )  e. 
_V )
37 canth2g 7664 . . . . . . . . 9  |-  ( ( A  \  { x } )  e.  _V  ->  ( A  \  {
x } )  ~<  ~P ( A  \  {
x } ) )
38 domunsn 7660 . . . . . . . . 9  |-  ( ( A  \  { x } )  ~<  ~P ( A  \  { x }
)  ->  ( ( A  \  { x }
)  u.  { x } )  ~<_  ~P ( A  \  { x }
) )
3936, 37, 383syl 20 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  u.  { x }
)  ~<_  ~P ( A  \  { x } ) )
4034, 39eqbrtrrd 4461 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~<_  ~P ( A  \  { x } ) )
41 xpdom1g 7607 . . . . . . 7  |-  ( ( ~P { x }  e.  _V  /\  A  ~<_  ~P ( A  \  {
x } ) )  ->  ( A  X.  ~P { x } )  ~<_  ( ~P ( A 
\  { x }
)  X.  ~P {
x } ) )
4228, 40, 41sylancr 661 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  ~P { x } )  ~<_  ( ~P ( A 
\  { x }
)  X.  ~P {
x } ) )
43 endomtr 7566 . . . . . 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 659 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ( ~P ( A  \  { x }
)  X.  ~P {
x } ) )
45 pwcdaen 8556 . . . . . . 7  |-  ( ( ( A  \  {
x } )  e. 
_V  /\  { x }  e.  _V )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ( ~P ( A  \  {
x } )  X. 
~P { x }
) )
4636, 27, 45sylancl 660 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ( ~P ( A  \  {
x } )  X. 
~P { x }
) )
4746ensymd 7559 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ~P ( A 
\  { x }
)  X.  ~P {
x } )  ~~  ~P ( ( A  \  { x } )  +c  { x }
) )
48 domentr 7567 . . . . 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 659 . . . 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 3677 . . . . . . . . 9  |-  ( ( A  \  { x } )  i^i  {
x } )  =  ( { x }  i^i  ( A  \  {
x } ) )
52 disjdif 3888 . . . . . . . . 9  |-  ( { x }  i^i  ( A  \  { x }
) )  =  (/)
5351, 52eqtri 2483 . . . . . . . 8  |-  ( ( A  \  { x } )  i^i  {
x } )  =  (/)
5453a1i 11 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  i^i  { x }
)  =  (/) )
55 cdaun 8543 . . . . . . 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 1226 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  +c  { x }
)  ~~  ( ( A  \  { x }
)  u.  { x } ) )
5756, 34breqtrd 4463 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  +c  { x }
)  ~~  A )
58 pwen 7683 . . . . 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 7567 . . . 4  |-  ( ( ( A  X.  2o )  ~<_  ~P ( ( A 
\  { x }
)  +c  { x } )  /\  ~P ( ( A  \  { x } )  +c  { x }
)  ~~  ~P A
)  ->  ( A  X.  2o )  ~<_  ~P A
)
6149, 59, 60syl2anc 659 . . 3  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ~P A )
629, 61exlimddv 1731 . 2  |-  ( 1o 
~<  A  ->  ( A  X.  2o )  ~<_  ~P A )
63 domtr 7561 . 2  |-  ( ( ( A  +c  2o )  ~<_  ( A  X.  2o )  /\  ( A  X.  2o )  ~<_  ~P A )  ->  ( A  +c  2o )  ~<_  ~P A )
643, 62, 63syl2anc 659 1  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   {cpr 4018   class class class wbr 4439    X. cxp 4986  (class class class)co 6270   1oc1o 7115   2oc2o 7116    ~~ cen 7506    ~<_ cdom 7507    ~< csdm 7508    +c ccda 8538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-1o 7122  df-2o 7123  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-cda 8539
This theorem is referenced by:  canthp1lem2  9020  canthp1  9021
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