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Theorem pwcdandom 8949
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 8947 . 2  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) )
2 df1o2 7045 . . . . . . 7  |-  1o  =  { (/) }
32xpeq2i 4972 . . . . . 6  |-  ( A  X.  1o )  =  ( A  X.  { (/)
} )
4 reldom 7429 . . . . . . . 8  |-  Rel  ~<_
54brrelex2i 4991 . . . . . . 7  |-  ( om  ~<_  A  ->  A  e.  _V )
6 0ex 4533 . . . . . . 7  |-  (/)  e.  _V
7 xpsneng 7509 . . . . . . 7  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
85, 6, 7sylancl 662 . . . . . 6  |-  ( om  ~<_  A  ->  ( A  X.  { (/) } )  ~~  A )
93, 8syl5eqbr 4436 . . . . 5  |-  ( om  ~<_  A  ->  ( A  X.  1o )  ~~  A
)
109ensymd 7473 . . . 4  |-  ( om  ~<_  A  ->  A  ~~  ( A  X.  1o ) )
11 omex 7964 . . . . . . 7  |-  om  e.  _V
12 ordom 6598 . . . . . . . 8  |-  Ord  om
13 1onn 7191 . . . . . . . 8  |-  1o  e.  om
14 ordelss 4846 . . . . . . . 8  |-  ( ( Ord  om  /\  1o  e.  om )  ->  1o  C_ 
om )
1512, 13, 14mp2an 672 . . . . . . 7  |-  1o  C_  om
16 ssdomg 7468 . . . . . . 7  |-  ( om  e.  _V  ->  ( 1o  C_  om  ->  1o  ~<_  om ) )
1711, 15, 16mp2 9 . . . . . 6  |-  1o  ~<_  om
18 domtr 7475 . . . . . 6  |-  ( ( 1o  ~<_  om  /\  om  ~<_  A )  ->  1o  ~<_  A )
1917, 18mpan 670 . . . . 5  |-  ( om  ~<_  A  ->  1o  ~<_  A )
20 xpdom2g 7520 . . . . 5  |-  ( ( A  e.  _V  /\  1o 
~<_  A )  ->  ( A  X.  1o )  ~<_  ( A  X.  A ) )
215, 19, 20syl2anc 661 . . . 4  |-  ( om  ~<_  A  ->  ( A  X.  1o )  ~<_  ( A  X.  A ) )
22 endomtr 7480 . . . 4  |-  ( ( A  ~~  ( A  X.  1o )  /\  ( A  X.  1o )  ~<_  ( A  X.  A ) )  ->  A  ~<_  ( A  X.  A ) )
2310, 21, 22syl2anc 661 . . 3  |-  ( om  ~<_  A  ->  A  ~<_  ( A  X.  A ) )
24 cdadom2 8471 . . 3  |-  ( A  ~<_  ( A  X.  A
)  ->  ( A  +c  A )  ~<_  ( A  +c  ( A  X.  A ) ) )
25 domtr 7475 . . . 4  |-  ( ( ~P A  ~<_  ( A  +c  A )  /\  ( A  +c  A
)  ~<_  ( A  +c  ( A  X.  A
) ) )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) )
2625expcom 435 . . 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 1758   _Vcvv 3078    C_ wss 3439   (/)c0 3748   ~Pcpw 3971   {csn 3988   class class class wbr 4403   Ord word 4829    X. cxp 4949  (class class class)co 6203   omcom 6589   1oc1o 7026    ~~ cen 7420    ~<_ cdom 7421    +c ccda 8451
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-seqom 7016  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-oexp 7039  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7839  df-har 7888  df-cnf 7983  df-card 8224  df-cda 8452
This theorem is referenced by:  gchcdaidm  8950
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