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Theorem gchxpidm 8500
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 4299 . . . . . . . 8  |-  (/)  e.  _V
21a1i 11 . . . . . . 7  |-  ( -.  A  e.  Fin  ->  (/)  e.  _V )
3 xpsneng 7152 . . . . . . 7  |-  ( ( A  e. GCH  /\  (/)  e.  _V )  ->  ( A  X.  { (/) } )  ~~  A )
42, 3sylan2 461 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  { (/)
} )  ~~  A
)
54ensymd 7117 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  X.  { (/) } ) )
6 df1o2 6695 . . . . . . 7  |-  1o  =  { (/) }
7 id 20 . . . . . . . . . . . 12  |-  ( A  =  (/)  ->  A  =  (/) )
8 0fin 7295 . . . . . . . . . . . 12  |-  (/)  e.  Fin
97, 8syl6eqel 2492 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  A  e. 
Fin )
109necon3bi 2608 . . . . . . . . . 10  |-  ( -.  A  e.  Fin  ->  A  =/=  (/) )
1110adantl 453 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  =/=  (/) )
12 0sdomg 7195 . . . . . . . . . 10  |-  ( A  e. GCH  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
1312adantr 452 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
1411, 13mpbird 224 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  -> 
(/)  ~<  A )
15 0sdom1dom 7265 . . . . . . . 8  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
1614, 15sylib 189 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  1o  ~<_  A )
176, 16syl5eqbrr 4206 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  { (/) }  ~<_  A )
18 xpdom2g 7163 . . . . . 6  |-  ( ( A  e. GCH  /\  { (/)
}  ~<_  A )  -> 
( A  X.  { (/)
} )  ~<_  ( A  X.  A ) )
1917, 18syldan 457 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  { (/)
} )  ~<_  ( A  X.  A ) )
20 endomtr 7124 . . . . 5  |-  ( ( A  ~~  ( A  X.  { (/) } )  /\  ( A  X.  { (/) } )  ~<_  ( A  X.  A ) )  ->  A  ~<_  ( A  X.  A ) )
215, 19, 20syl2anc 643 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  X.  A ) )
22 canth2g 7220 . . . . . . . . . 10  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
2322adantr 452 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<  ~P A
)
24 sdomdom 7094 . . . . . . . . 9  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
2523, 24syl 16 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ~P A )
26 xpdom1g 7164 . . . . . . . 8  |-  ( ( A  e. GCH  /\  A  ~<_  ~P A )  ->  ( A  X.  A )  ~<_  ( ~P A  X.  A
) )
2725, 26syldan 457 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ( ~P A  X.  A ) )
28 pwexg 4343 . . . . . . . . 9  |-  ( A  e. GCH  ->  ~P A  e. 
_V )
2928adantr 452 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P A  e.  _V )
30 xpdom2g 7163 . . . . . . . 8  |-  ( ( ~P A  e.  _V  /\  A  ~<_  ~P A )  -> 
( ~P A  X.  A )  ~<_  ( ~P A  X.  ~P A
) )
3129, 25, 30syl2anc 643 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  A )  ~<_  ( ~P A  X.  ~P A
) )
32 domtr 7119 . . . . . . 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
) )
3327, 31, 32syl2anc 643 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ( ~P A  X.  ~P A ) )
34 simpl 444 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  e. GCH )
35 pwcdaen 8021 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A ) )
3634, 35syldan 457 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A
) )
3736ensymd 7117 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  ~P A )  ~~  ~P ( A  +c  A
) )
38 gchcdaidm 8499 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
39 pwen 7239 . . . . . . . 8  |-  ( ( A  +c  A ) 
~~  A  ->  ~P ( A  +c  A
)  ~~  ~P A
)
4038, 39syl 16 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  A )  ~~  ~P A )
41 entr 7118 . . . . . . 7  |-  ( ( ( ~P A  X.  ~P A )  ~~  ~P ( A  +c  A
)  /\  ~P ( A  +c  A )  ~~  ~P A )  ->  ( ~P A  X.  ~P A
)  ~~  ~P A
)
4237, 40, 41syl2anc 643 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  ~P A )  ~~  ~P A )
43 domentr 7125 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( ~P A  X.  ~P A )  /\  ( ~P A  X.  ~P A )  ~~  ~P A )  ->  ( A  X.  A )  ~<_  ~P A )
4433, 42, 43syl2anc 643 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ~P A )
45 gchinf 8488 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  om  ~<_  A )
46 pwxpndom 8497 . . . . . . 7  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  X.  A
) )
4745, 46syl 16 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ~P A  ~<_  ( A  X.  A ) )
48 ensym 7115 . . . . . . 7  |-  ( ( A  X.  A ) 
~~  ~P A  ->  ~P A  ~~  ( A  X.  A ) )
49 endom 7093 . . . . . . 7  |-  ( ~P A  ~~  ( A  X.  A )  ->  ~P A  ~<_  ( A  X.  A ) )
5048, 49syl 16 . . . . . 6  |-  ( ( A  X.  A ) 
~~  ~P A  ->  ~P A  ~<_  ( A  X.  A ) )
5147, 50nsyl 115 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ( A  X.  A )  ~~  ~P A )
52 brsdom 7089 . . . . 5  |-  ( ( A  X.  A ) 
~<  ~P A  <->  ( ( A  X.  A )  ~<_  ~P A  /\  -.  ( A  X.  A )  ~~  ~P A ) )
5344, 51, 52sylanbrc 646 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<  ~P A )
5421, 53jca 519 . . 3  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  ~<_  ( A  X.  A )  /\  ( A  X.  A
)  ~<  ~P A ) )
55 gchen1 8456 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  ( A  X.  A )  /\  ( A  X.  A )  ~<  ~P A
) )  ->  A  ~~  ( A  X.  A
) )
5654, 55mpdan 650 . 2  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  X.  A ) )
5756ensymd 7117 1  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~~  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916   (/)c0 3588   ~Pcpw 3759   {csn 3774   class class class wbr 4172   omcom 4804    X. cxp 4835  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067   Fincfn 7068    +c ccda 8003  GCHcgch 8451
This theorem is referenced by:  gchhar  8502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-seqom 6664  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-oexp 6689  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-cnf 7573  df-card 7782  df-cda 8004  df-fin4 8123  df-gch 8452
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