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Theorem pwcdandom 8289
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 8287 . 2  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) )
2 df1o2 6491 . . . . . . . 8  |-  1o  =  { (/) }
32xpeq2i 4710 . . . . . . 7  |-  ( A  X.  1o )  =  ( A  X.  { (/)
} )
4 reldom 6869 . . . . . . . . 9  |-  Rel  ~<_
54brrelex2i 4730 . . . . . . . 8  |-  ( om  ~<_  A  ->  A  e.  _V )
6 0ex 4150 . . . . . . . 8  |-  (/)  e.  _V
7 xpsneng 6947 . . . . . . . 8  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
85, 6, 7sylancl 643 . . . . . . 7  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  A )
93, 8syl5eqbr 4056 . . . . . 6  |-  ( om  ~<_  A  ->  ( A  X.  1o )  ~~  A
)
10 ensym 6910 . . . . . 6  |-  ( ( A  X.  1o ) 
~~  A  ->  A  ~~  ( A  X.  1o ) )
119, 10syl 15 . . . . 5  |-  ( om  ~<_  A  ->  A  ~~  ( A  X.  1o ) )
12 omex 7344 . . . . . . . 8  |-  om  e.  _V
13 ordom 4665 . . . . . . . . 9  |-  Ord  om
14 1onn 6637 . . . . . . . . 9  |-  1o  e.  om
15 ordelss 4408 . . . . . . . . 9  |-  ( ( Ord  om  /\  1o  e.  om )  ->  1o  C_ 
om )
1613, 14, 15mp2an 653 . . . . . . . 8  |-  1o  C_  om
17 ssdomg 6907 . . . . . . . 8  |-  ( om  e.  _V  ->  ( 1o  C_  om  ->  1o  ~<_  om ) )
1812, 16, 17mp2 17 . . . . . . 7  |-  1o  ~<_  om
19 domtr 6914 . . . . . . 7  |-  ( ( 1o  ~<_  om  /\  om  ~<_  A )  ->  1o  ~<_  A )
2018, 19mpan 651 . . . . . 6  |-  ( om  ~<_  A  ->  1o  ~<_  A )
21 xpdom2g 6958 . . . . . 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 6919 . . . . 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 7813 . . . 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 6914 . . . 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 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023   Ord word 4391   omcom 4656    X. cxp 4687  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861    +c ccda 7793
This theorem is referenced by:  gchcdaidm  8290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-cnf 7363  df-card 7572  df-cda 7794
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