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Theorem canthp1lem1 8906
Description: Lemma for canthp1 8908. (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 7598 . . 3  |-  1o  ~<  2o
2 cdaxpdom 8445 . . 3  |-  ( ( 1o  ~<  A  /\  1o  ~<  2o )  -> 
( A  +c  2o )  ~<_  ( A  X.  2o ) )
31, 2mpan2 671 . 2  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ( A  X.  2o ) )
4 sdom0 7529 . . . . . 6  |-  -.  1o  ~< 
(/)
5 breq2 4380 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
~<  A  <->  1o  ~<  (/) ) )
64, 5mtbiri 303 . . . . 5  |-  ( A  =  (/)  ->  -.  1o  ~<  A )
76con2i 120 . . . 4  |-  ( 1o 
~<  A  ->  -.  A  =  (/) )
8 neq0 3731 . . . 4  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
97, 8sylib 196 . . 3  |-  ( 1o 
~<  A  ->  E. x  x  e.  A )
10 relsdom 7403 . . . . . . . . . 10  |-  Rel  ~<
1110brrelex2i 4964 . . . . . . . . 9  |-  ( 1o 
~<  A  ->  A  e. 
_V )
1211adantr 465 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  e.  _V )
13 enrefg 7427 . . . . . . . 8  |-  ( A  e.  _V  ->  A  ~~  A )
1412, 13syl 16 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~~  A )
15 df2o2 7020 . . . . . . . . 9  |-  2o  =  { (/) ,  { (/) } }
16 pwpw0 4105 . . . . . . . . 9  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
1715, 16eqtr4i 2481 . . . . . . . 8  |-  2o  =  ~P { (/) }
18 0ex 4506 . . . . . . . . . 10  |-  (/)  e.  _V
19 vex 3057 . . . . . . . . . 10  |-  x  e. 
_V
20 en2sn 7475 . . . . . . . . . 10  |-  ( (
(/)  e.  _V  /\  x  e.  _V )  ->  { (/) } 
~~  { x }
)
2118, 19, 20mp2an 672 . . . . . . . . 9  |-  { (/) } 
~~  { x }
22 pwen 7570 . . . . . . . . 9  |-  ( {
(/) }  ~~  { x }  ->  ~P { (/) } 
~~  ~P { x }
)
2321, 22ax-mp 5 . . . . . . . 8  |-  ~P { (/)
}  ~~  ~P { x }
2417, 23eqbrtri 4395 . . . . . . 7  |-  2o  ~~  ~P { x }
25 xpen 7560 . . . . . . 7  |-  ( ( A  ~~  A  /\  2o  ~~  ~P { x } )  ->  ( A  X.  2o )  ~~  ( A  X.  ~P {
x } ) )
2614, 24, 25sylancl 662 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~~  ( A  X.  ~P { x } ) )
27 snex 4617 . . . . . . . 8  |-  { x }  e.  _V
2827pwex 4559 . . . . . . 7  |-  ~P {
x }  e.  _V
29 uncom 3584 . . . . . . . . 9  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( { x }  u.  ( A  \  {
x } ) )
30 simpr 461 . . . . . . . . . . 11  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  x  e.  A )
3130snssd 4102 . . . . . . . . . 10  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  { x }  C_  A )
32 undif 3843 . . . . . . . . . 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 2502 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  u.  { x }
)  =  A )
35 difexg 4524 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  \  { x }
)  e.  _V )
3612, 35syl 16 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  \  {
x } )  e. 
_V )
37 canth2g 7551 . . . . . . . . 9  |-  ( ( A  \  { x } )  e.  _V  ->  ( A  \  {
x } )  ~<  ~P ( A  \  {
x } ) )
38 domunsn 7547 . . . . . . . . 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 4398 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  A  ~<_  ~P ( A  \  { x } ) )
41 xpdom1g 7494 . . . . . . 7  |-  ( ( ~P { x }  e.  _V  /\  A  ~<_  ~P ( A  \  {
x } ) )  ->  ( A  X.  ~P { x } )  ~<_  ( ~P ( A 
\  { x }
)  X.  ~P {
x } ) )
4228, 40, 41sylancr 663 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  ~P { x } )  ~<_  ( ~P ( A 
\  { x }
)  X.  ~P {
x } ) )
43 endomtr 7453 . . . . . 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 661 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ( ~P ( A  \  { x }
)  X.  ~P {
x } ) )
45 pwcdaen 8441 . . . . . . 7  |-  ( ( ( A  \  {
x } )  e. 
_V  /\  { x }  e.  _V )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ( ~P ( A  \  {
x } )  X. 
~P { x }
) )
4636, 27, 45sylancl 662 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ~P ( ( A 
\  { x }
)  +c  { x } )  ~~  ( ~P ( A  \  {
x } )  X. 
~P { x }
) )
4746ensymd 7446 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ~P ( A 
\  { x }
)  X.  ~P {
x } )  ~~  ~P ( ( A  \  { x } )  +c  { x }
) )
48 domentr 7454 . . . . 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 661 . . . 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 3627 . . . . . . . . 9  |-  ( ( A  \  { x } )  i^i  {
x } )  =  ( { x }  i^i  ( A  \  {
x } ) )
52 disjdif 3835 . . . . . . . . 9  |-  ( { x }  i^i  ( A  \  { x }
) )  =  (/)
5351, 52eqtri 2478 . . . . . . . 8  |-  ( ( A  \  { x } )  i^i  {
x } )  =  (/)
5453a1i 11 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  i^i  { x }
)  =  (/) )
55 cdaun 8428 . . . . . . 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 1219 . . . . . 6  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  +c  { x }
)  ~~  ( ( A  \  { x }
)  u.  { x } ) )
5756, 34breqtrd 4400 . . . . 5  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( ( A  \  { x } )  +c  { x }
)  ~~  A )
58 pwen 7570 . . . . 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 7454 . . . 4  |-  ( ( ( A  X.  2o )  ~<_  ~P ( ( A 
\  { x }
)  +c  { x } )  /\  ~P ( ( A  \  { x } )  +c  { x }
)  ~~  ~P A
)  ->  ( A  X.  2o )  ~<_  ~P A
)
6149, 59, 60syl2anc 661 . . 3  |-  ( ( 1o  ~<  A  /\  x  e.  A )  ->  ( A  X.  2o )  ~<_  ~P A )
629, 61exlimddv 1693 . 2  |-  ( 1o 
~<  A  ->  ( A  X.  2o )  ~<_  ~P A )
63 domtr 7448 . 2  |-  ( ( ( A  +c  2o )  ~<_  ( A  X.  2o )  /\  ( A  X.  2o )  ~<_  ~P A )  ->  ( A  +c  2o )  ~<_  ~P A )
643, 62, 63syl2anc 661 1  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1757   _Vcvv 3054    \ cdif 3409    u. cun 3410    i^i cin 3411    C_ wss 3412   (/)c0 3721   ~Pcpw 3944   {csn 3961   {cpr 3963   class class class wbr 4376    X. cxp 4922  (class class class)co 6176   1oc1o 6999   2oc2o 7000    ~~ cen 7393    ~<_ cdom 7394    ~< csdm 7395    +c ccda 8423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-1o 7006  df-2o 7007  df-er 7187  df-map 7302  df-en 7397  df-dom 7398  df-sdom 7399  df-cda 8424
This theorem is referenced by:  canthp1lem2  8907  canthp1  8908
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