MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwxpndom Structured version   Unicode version

Theorem pwxpndom 8572
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 8571 . 2  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  ( A  X.  A ) ) )
2 reldom 7144 . . . . . . 7  |-  Rel  ~<_
32brrelex2i 4948 . . . . . 6  |-  ( om  ~<_  A  ->  A  e.  _V )
4 xpexg 5018 . . . . . 6  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
53, 3, 4syl2anc 644 . . . . 5  |-  ( om  ~<_  A  ->  ( A  X.  A )  e.  _V )
6 cdadom3 8099 . . . . 5  |-  ( ( ( A  X.  A
)  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  ~<_  ( ( A  X.  A )  +c  A ) )
75, 3, 6syl2anc 644 . . . 4  |-  ( om  ~<_  A  ->  ( A  X.  A )  ~<_  ( ( A  X.  A )  +c  A ) )
8 cdacomen 8092 . . . 4  |-  ( ( A  X.  A )  +c  A )  ~~  ( A  +c  ( A  X.  A ) )
9 domentr 7195 . . . 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 645 . . 3  |-  ( om  ~<_  A  ->  ( A  X.  A )  ~<_  ( A  +c  ( A  X.  A ) ) )
11 domtr 7189 . . . 4  |-  ( ( ~P A  ~<_  ( A  X.  A )  /\  ( A  X.  A
)  ~<_  ( A  +c  ( A  X.  A
) ) )  ->  ~P A  ~<_  ( A  +c  ( A  X.  A
) ) )
1211expcom 426 . . 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 171 1  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1727   _Vcvv 2962   ~Pcpw 3823   class class class wbr 4237   omcom 4874    X. cxp 4905  (class class class)co 6110    ~~ cen 7135    ~<_ cdom 7136    +c ccda 8078
This theorem is referenced by:  gchxpidm  8575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-seqom 6734  df-1o 6753  df-2o 6754  df-oadd 6757  df-omul 6758  df-oexp 6759  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-oi 7508  df-har 7555  df-cnf 7646  df-card 7857  df-cda 8079
  Copyright terms: Public domain W3C validator