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Theorem gchxpidm 8171
Description: An infinite GCH-set is idempotent under cardinal product. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
gchxpidm  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~~  A )

Proof of Theorem gchxpidm
StepHypRef Expression
1 0ex 4047 . . . . . . . 8  |-  (/)  e.  _V
21a1i 12 . . . . . . 7  |-  ( -.  A  e.  Fin  ->  (/)  e.  _V )
3 xpsneng 6832 . . . . . . 7  |-  ( ( A  e. GCH  /\  (/)  e.  _V )  ->  ( A  X.  { (/) } )  ~~  A )
42, 3sylan2 462 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  { (/)
} )  ~~  A
)
5 ensym 6796 . . . . . 6  |-  ( ( A  X.  { (/) } )  ~~  A  ->  A  ~~  ( A  X.  { (/) } ) )
64, 5syl 17 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  X.  { (/) } ) )
7 df1o2 6377 . . . . . . 7  |-  1o  =  { (/) }
8 id 21 . . . . . . . . . . . 12  |-  ( A  =  (/)  ->  A  =  (/) )
9 0fin 6972 . . . . . . . . . . . 12  |-  (/)  e.  Fin
108, 9syl6eqel 2341 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  A  e. 
Fin )
1110necon3bi 2453 . . . . . . . . . 10  |-  ( -.  A  e.  Fin  ->  A  =/=  (/) )
1211adantl 454 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  =/=  (/) )
13 0sdomg 6875 . . . . . . . . . 10  |-  ( A  e. GCH  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
1413adantr 453 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
1512, 14mpbird 225 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  -> 
(/)  ~<  A )
16 0sdom1dom 6945 . . . . . . . 8  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
1715, 16sylib 190 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  1o  ~<_  A )
187, 17syl5eqbrr 3954 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  { (/) }  ~<_  A )
19 xpdom2g 6843 . . . . . 6  |-  ( ( A  e. GCH  /\  { (/)
}  ~<_  A )  -> 
( A  X.  { (/)
} )  ~<_  ( A  X.  A ) )
2018, 19syldan 458 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  { (/)
} )  ~<_  ( A  X.  A ) )
21 endomtr 6804 . . . . 5  |-  ( ( A  ~~  ( A  X.  { (/) } )  /\  ( A  X.  { (/) } )  ~<_  ( A  X.  A ) )  ->  A  ~<_  ( A  X.  A ) )
226, 20, 21syl2anc 645 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  X.  A ) )
23 canth2g 6900 . . . . . . . . . 10  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
2423adantr 453 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<  ~P A
)
25 sdomdom 6775 . . . . . . . . 9  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
2624, 25syl 17 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ~P A )
27 xpdom1g 6844 . . . . . . . 8  |-  ( ( A  e. GCH  /\  A  ~<_  ~P A )  ->  ( A  X.  A )  ~<_  ( ~P A  X.  A
) )
2826, 27syldan 458 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ( ~P A  X.  A ) )
29 pwexg 4088 . . . . . . . . 9  |-  ( A  e. GCH  ->  ~P A  e. 
_V )
3029adantr 453 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P A  e.  _V )
31 xpdom2g 6843 . . . . . . . 8  |-  ( ( ~P A  e.  _V  /\  A  ~<_  ~P A )  -> 
( ~P A  X.  A )  ~<_  ( ~P A  X.  ~P A
) )
3230, 26, 31syl2anc 645 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  A )  ~<_  ( ~P A  X.  ~P A
) )
33 domtr 6799 . . . . . . 7  |-  ( ( ( A  X.  A
)  ~<_  ( ~P A  X.  A )  /\  ( ~P A  X.  A
)  ~<_  ( ~P A  X.  ~P A ) )  ->  ( A  X.  A )  ~<_  ( ~P A  X.  ~P A
) )
3428, 32, 33syl2anc 645 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ( ~P A  X.  ~P A ) )
35 simpl 445 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  e. GCH )
36 pwcdaen 7695 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A ) )
3735, 36syldan 458 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A
) )
38 ensym 6796 . . . . . . . 8  |-  ( ~P ( A  +c  A
)  ~~  ( ~P A  X.  ~P A )  ->  ( ~P A  X.  ~P A )  ~~  ~P ( A  +c  A
) )
3937, 38syl 17 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  ~P A )  ~~  ~P ( A  +c  A
) )
40 gchcdaidm 8170 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
41 pwen 6919 . . . . . . . 8  |-  ( ( A  +c  A ) 
~~  A  ->  ~P ( A  +c  A
)  ~~  ~P A
)
4240, 41syl 17 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  A )  ~~  ~P A )
43 entr 6798 . . . . . . 7  |-  ( ( ( ~P A  X.  ~P A )  ~~  ~P ( A  +c  A
)  /\  ~P ( A  +c  A )  ~~  ~P A )  ->  ( ~P A  X.  ~P A
)  ~~  ~P A
)
4439, 42, 43syl2anc 645 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  ~P A )  ~~  ~P A )
45 domentr 6805 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( ~P A  X.  ~P A )  /\  ( ~P A  X.  ~P A )  ~~  ~P A )  ->  ( A  X.  A )  ~<_  ~P A )
4634, 44, 45syl2anc 645 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ~P A )
47 gchinf 8159 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  om  ~<_  A )
48 pwxpndom 8168 . . . . . . 7  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  X.  A
) )
4947, 48syl 17 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ~P A  ~<_  ( A  X.  A ) )
50 ensym 6796 . . . . . . 7  |-  ( ( A  X.  A ) 
~~  ~P A  ->  ~P A  ~~  ( A  X.  A ) )
51 endom 6774 . . . . . . 7  |-  ( ~P A  ~~  ( A  X.  A )  ->  ~P A  ~<_  ( A  X.  A ) )
5250, 51syl 17 . . . . . 6  |-  ( ( A  X.  A ) 
~~  ~P A  ->  ~P A  ~<_  ( A  X.  A ) )
5349, 52nsyl 115 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ( A  X.  A )  ~~  ~P A )
54 brsdom 6770 . . . . 5  |-  ( ( A  X.  A ) 
~<  ~P A  <->  ( ( A  X.  A )  ~<_  ~P A  /\  -.  ( A  X.  A )  ~~  ~P A ) )
5546, 53, 54sylanbrc 648 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<  ~P A )
5622, 55jca 520 . . 3  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  ~<_  ( A  X.  A )  /\  ( A  X.  A
)  ~<  ~P A ) )
57 gchen1 8127 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  ( A  X.  A )  /\  ( A  X.  A )  ~<  ~P A
) )  ->  A  ~~  ( A  X.  A
) )
5856, 57mpdan 652 . 2  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  X.  A ) )
59 ensym 6796 . 2  |-  ( A 
~~  ( A  X.  A )  ->  ( A  X.  A )  ~~  A )
6058, 59syl 17 1  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~~  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   _Vcvv 2727   (/)c0 3362   ~Pcpw 3530   {csn 3544   class class class wbr 3920   omcom 4547    X. cxp 4578  (class class class)co 5710   1oc1o 6358    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748   Fincfn 6749    +c ccda 7677  GCHcgch 8122
This theorem is referenced by:  gchhar  8173
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-seqom 6346  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-oexp 6371  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-har 7156  df-cnf 7247  df-card 7456  df-cda 7678  df-fin4 7797  df-gch 8123
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