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Theorem pwcdandom 8305
Description: The powerset of a Dedekind-infinite set does not inject into its cardinal sum with itself. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
pwcdandom  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  A
) )

Proof of Theorem pwcdandom
StepHypRef Expression
1 pwxpndom2 8303 . 2  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) )
2 df1o2 6507 . . . . . . . 8  |-  1o  =  { (/) }
32xpeq2i 4726 . . . . . . 7  |-  ( A  X.  1o )  =  ( A  X.  { (/)
} )
4 reldom 6885 . . . . . . . . 9  |-  Rel  ~<_
54brrelex2i 4746 . . . . . . . 8  |-  ( om  ~<_  A  ->  A  e.  _V )
6 0ex 4166 . . . . . . . 8  |-  (/)  e.  _V
7 xpsneng 6963 . . . . . . . 8  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
85, 6, 7sylancl 643 . . . . . . 7  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  A )
93, 8syl5eqbr 4072 . . . . . 6  |-  ( om  ~<_  A  ->  ( A  X.  1o )  ~~  A
)
10 ensym 6926 . . . . . 6  |-  ( ( A  X.  1o ) 
~~  A  ->  A  ~~  ( A  X.  1o ) )
119, 10syl 15 . . . . 5  |-  ( om  ~<_  A  ->  A  ~~  ( A  X.  1o ) )
12 omex 7360 . . . . . . . 8  |-  om  e.  _V
13 ordom 4681 . . . . . . . . 9  |-  Ord  om
14 1onn 6653 . . . . . . . . 9  |-  1o  e.  om
15 ordelss 4424 . . . . . . . . 9  |-  ( ( Ord  om  /\  1o  e.  om )  ->  1o  C_ 
om )
1613, 14, 15mp2an 653 . . . . . . . 8  |-  1o  C_  om
17 ssdomg 6923 . . . . . . . 8  |-  ( om  e.  _V  ->  ( 1o  C_  om  ->  1o  ~<_  om ) )
1812, 16, 17mp2 17 . . . . . . 7  |-  1o  ~<_  om
19 domtr 6930 . . . . . . 7  |-  ( ( 1o  ~<_  om  /\  om  ~<_  A )  ->  1o  ~<_  A )
2018, 19mpan 651 . . . . . 6  |-  ( om  ~<_  A  ->  1o  ~<_  A )
21 xpdom2g 6974 . . . . . 6  |-  ( ( A  e.  _V  /\  1o 
~<_  A )  ->  ( A  X.  1o )  ~<_  ( A  X.  A ) )
225, 20, 21syl2anc 642 . . . . 5  |-  ( om  ~<_  A  ->  ( A  X.  1o )  ~<_  ( A  X.  A ) )
23 endomtr 6935 . . . . 5  |-  ( ( A  ~~  ( A  X.  1o )  /\  ( A  X.  1o )  ~<_  ( A  X.  A ) )  ->  A  ~<_  ( A  X.  A ) )
2411, 22, 23syl2anc 642 . . . 4  |-  ( om  ~<_  A  ->  A  ~<_  ( A  X.  A ) )
25 cdadom2 7829 . . . 4  |-  ( A  ~<_  ( A  X.  A
)  ->  ( A  +c  A )  ~<_  ( A  +c  ( A  X.  A ) ) )
2624, 25syl 15 . . 3  |-  ( om  ~<_  A  ->  ( A  +c  A )  ~<_  ( A  +c  ( A  X.  A ) ) )
27 domtr 6930 . . . 4  |-  ( ( ~P A  ~<_  ( A  +c  A )  /\  ( A  +c  A
)  ~<_  ( A  +c  ( A  X.  A
) ) )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) )
2827expcom 424 . . 3  |-  ( ( A  +c  A )  ~<_  ( A  +c  ( A  X.  A ) )  ->  ( ~P A  ~<_  ( A  +c  A
)  ->  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) ) )
2926, 28syl 15 . 2  |-  ( om  ~<_  A  ->  ( ~P A  ~<_  ( A  +c  A )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) ) )
301, 29mtod 168 1  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   class class class wbr 4039   Ord word 4407   omcom 4672    X. cxp 4703  (class class class)co 5874   1oc1o 6488    ~~ cen 6876    ~<_ cdom 6877    +c ccda 7809
This theorem is referenced by:  gchcdaidm  8306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-oexp 6501  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-cnf 7379  df-card 7588  df-cda 7810
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