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Theorem pwxpndom 9047
Description: The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
pwxpndom  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  X.  A
) )

Proof of Theorem pwxpndom
StepHypRef Expression
1 pwxpndom2 9046 . 2  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) )
2 reldom 7524 . . . . . . 7  |-  Rel  ~<_
32brrelex2i 5031 . . . . . 6  |-  ( om  ~<_  A  ->  A  e.  _V )
4 xpexg 6587 . . . . . 6  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
53, 3, 4syl2anc 661 . . . . 5  |-  ( om  ~<_  A  ->  ( A  X.  A )  e.  _V )
6 cdadom3 8571 . . . . 5  |-  ( ( ( A  X.  A
)  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  ~<_  ( ( A  X.  A )  +c  A ) )
75, 3, 6syl2anc 661 . . . 4  |-  ( om  ~<_  A  ->  ( A  X.  A )  ~<_  ( ( A  X.  A )  +c  A ) )
8 cdacomen 8564 . . . 4  |-  ( ( A  X.  A )  +c  A )  ~~  ( A  +c  ( A  X.  A ) )
9 domentr 7576 . . . 4  |-  ( ( ( A  X.  A
)  ~<_  ( ( A  X.  A )  +c  A )  /\  (
( A  X.  A
)  +c  A ) 
~~  ( A  +c  ( A  X.  A
) ) )  -> 
( A  X.  A
)  ~<_  ( A  +c  ( A  X.  A
) ) )
107, 8, 9sylancl 662 . . 3  |-  ( om  ~<_  A  ->  ( A  X.  A )  ~<_  ( A  +c  ( A  X.  A ) ) )
11 domtr 7570 . . . 4  |-  ( ( ~P A  ~<_  ( A  X.  A )  /\  ( A  X.  A
)  ~<_  ( A  +c  ( A  X.  A
) ) )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) )
1211expcom 435 . . 3  |-  ( ( A  X.  A )  ~<_  ( A  +c  ( A  X.  A ) )  ->  ( ~P A  ~<_  ( A  X.  A
)  ->  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) ) )
1310, 12syl 16 . 2  |-  ( om  ~<_  A  ->  ( ~P A  ~<_  ( A  X.  A )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) ) )
141, 13mtod 177 1  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1804   _Vcvv 3095   ~Pcpw 3997   class class class wbr 4437    X. cxp 4987  (class class class)co 6281   omcom 6685    ~~ cen 7515    ~<_ cdom 7516    +c ccda 8550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-seqom 7115  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-oexp 7138  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-har 7987  df-cnf 8082  df-card 8323  df-cda 8551
This theorem is referenced by:  gchxpidm  9050
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