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Theorem pwcdandom 9034
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 9032 . 2  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) )
2 df1o2 7134 . . . . . . 7  |-  1o  =  { (/) }
32xpeq2i 5009 . . . . . 6  |-  ( A  X.  1o )  =  ( A  X.  { (/)
} )
4 reldom 7515 . . . . . . . 8  |-  Rel  ~<_
54brrelex2i 5030 . . . . . . 7  |-  ( om  ~<_  A  ->  A  e.  _V )
6 0ex 4569 . . . . . . 7  |-  (/)  e.  _V
7 xpsneng 7595 . . . . . . 7  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
85, 6, 7sylancl 660 . . . . . 6  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  A )
93, 8syl5eqbr 4472 . . . . 5  |-  ( om  ~<_  A  ->  ( A  X.  1o )  ~~  A
)
109ensymd 7559 . . . 4  |-  ( om  ~<_  A  ->  A  ~~  ( A  X.  1o ) )
11 omex 8051 . . . . . . 7  |-  om  e.  _V
12 ordom 6682 . . . . . . . 8  |-  Ord  om
13 1onn 7280 . . . . . . . 8  |-  1o  e.  om
14 ordelss 4883 . . . . . . . 8  |-  ( ( Ord  om  /\  1o  e.  om )  ->  1o  C_ 
om )
1512, 13, 14mp2an 670 . . . . . . 7  |-  1o  C_  om
16 ssdomg 7554 . . . . . . 7  |-  ( om  e.  _V  ->  ( 1o  C_  om  ->  1o  ~<_  om ) )
1711, 15, 16mp2 9 . . . . . 6  |-  1o  ~<_  om
18 domtr 7561 . . . . . 6  |-  ( ( 1o  ~<_  om  /\  om  ~<_  A )  ->  1o  ~<_  A )
1917, 18mpan 668 . . . . 5  |-  ( om  ~<_  A  ->  1o  ~<_  A )
20 xpdom2g 7606 . . . . 5  |-  ( ( A  e.  _V  /\  1o 
~<_  A )  ->  ( A  X.  1o )  ~<_  ( A  X.  A ) )
215, 19, 20syl2anc 659 . . . 4  |-  ( om  ~<_  A  ->  ( A  X.  1o )  ~<_  ( A  X.  A ) )
22 endomtr 7566 . . . 4  |-  ( ( A  ~~  ( A  X.  1o )  /\  ( A  X.  1o )  ~<_  ( A  X.  A ) )  ->  A  ~<_  ( A  X.  A ) )
2310, 21, 22syl2anc 659 . . 3  |-  ( om  ~<_  A  ->  A  ~<_  ( A  X.  A ) )
24 cdadom2 8558 . . 3  |-  ( A  ~<_  ( A  X.  A
)  ->  ( A  +c  A )  ~<_  ( A  +c  ( A  X.  A ) ) )
25 domtr 7561 . . . 4  |-  ( ( ~P A  ~<_  ( A  +c  A )  /\  ( A  +c  A
)  ~<_  ( A  +c  ( A  X.  A
) ) )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) )
2625expcom 433 . . 3  |-  ( ( A  +c  A )  ~<_  ( A  +c  ( A  X.  A ) )  ->  ( ~P A  ~<_  ( A  +c  A
)  ->  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) ) )
2723, 24, 263syl 20 . 2  |-  ( om  ~<_  A  ->  ( ~P A  ~<_  ( A  +c  A )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) ) )
281, 27mtod 177 1  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1823   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   class class class wbr 4439   Ord word 4866    X. cxp 4986  (class class class)co 6270   omcom 6673   1oc1o 7115    ~~ cen 7506    ~<_ cdom 7507    +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-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  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-reu 2811  df-rmo 2812  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-se 4828  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-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-oexp 7128  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-har 7976  df-cnf 8070  df-card 8311  df-cda 8539
This theorem is referenced by:  gchcdaidm  9035
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