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Theorem pwxpndom 8375
Description: The powerset of a Dedekind-infinite set does not inject into its cross 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 8374 . 2  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) )
2 reldom 6954 . . . . . . 7  |-  Rel  ~<_
32brrelex2i 4809 . . . . . 6  |-  ( om  ~<_  A  ->  A  e.  _V )
4 xpexg 4879 . . . . . 6  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
53, 3, 4syl2anc 642 . . . . 5  |-  ( om  ~<_  A  ->  ( A  X.  A )  e.  _V )
6 cdadom3 7901 . . . . 5  |-  ( ( ( A  X.  A
)  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  ~<_  ( ( A  X.  A )  +c  A ) )
75, 3, 6syl2anc 642 . . . 4  |-  ( om  ~<_  A  ->  ( A  X.  A )  ~<_  ( ( A  X.  A )  +c  A ) )
8 cdacomen 7894 . . . 4  |-  ( ( A  X.  A )  +c  A )  ~~  ( A  +c  ( A  X.  A ) )
9 domentr 7005 . . . 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 643 . . 3  |-  ( om  ~<_  A  ->  ( A  X.  A )  ~<_  ( A  +c  ( A  X.  A ) ) )
11 domtr 6999 . . . 4  |-  ( ( ~P A  ~<_  ( A  X.  A )  /\  ( A  X.  A
)  ~<_  ( A  +c  ( A  X.  A
) ) )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) )
1211expcom 424 . . 3  |-  ( ( A  X.  A )  ~<_  ( A  +c  ( A  X.  A ) )  ->  ( ~P A  ~<_  ( A  X.  A
)  ->  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) ) )
1310, 12syl 15 . 2  |-  ( om  ~<_  A  ->  ( ~P A  ~<_  ( A  X.  A )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) ) )
141, 13mtod 168 1  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1710   _Vcvv 2864   ~Pcpw 3701   class class class wbr 4102   omcom 4735    X. cxp 4766  (class class class)co 5942    ~~ cen 6945    ~<_ cdom 6946    +c ccda 7880
This theorem is referenced by:  gchxpidm  8378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-seqom 6544  df-1o 6563  df-2o 6564  df-oadd 6567  df-omul 6568  df-oexp 6569  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-oi 7312  df-har 7359  df-cnf 7450  df-card 7659  df-cda 7881
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