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Theorem gchxpidm 8857
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 4443 . . . . . . . 8  |-  (/)  e.  _V
21a1i 11 . . . . . . 7  |-  ( -.  A  e.  Fin  ->  (/)  e.  _V )
3 xpsneng 7417 . . . . . . 7  |-  ( ( A  e. GCH  /\  (/)  e.  _V )  ->  ( A  X.  { (/) } )  ~~  A )
42, 3sylan2 474 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  { (/)
} )  ~~  A
)
54ensymd 7381 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  X.  { (/) } ) )
6 df1o2 6953 . . . . . . 7  |-  1o  =  { (/) }
7 id 22 . . . . . . . . . . . 12  |-  ( A  =  (/)  ->  A  =  (/) )
8 0fin 7561 . . . . . . . . . . . 12  |-  (/)  e.  Fin
97, 8syl6eqel 2531 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  A  e. 
Fin )
109necon3bi 2676 . . . . . . . . . 10  |-  ( -.  A  e.  Fin  ->  A  =/=  (/) )
1110adantl 466 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  =/=  (/) )
12 0sdomg 7461 . . . . . . . . . 10  |-  ( A  e. GCH  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
1312adantr 465 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
1411, 13mpbird 232 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  -> 
(/)  ~<  A )
15 0sdom1dom 7531 . . . . . . . 8  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
1614, 15sylib 196 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  1o  ~<_  A )
176, 16syl5eqbrr 4347 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  { (/) }  ~<_  A )
18 xpdom2g 7428 . . . . . 6  |-  ( ( A  e. GCH  /\  { (/)
}  ~<_  A )  -> 
( A  X.  { (/)
} )  ~<_  ( A  X.  A ) )
1917, 18syldan 470 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  { (/)
} )  ~<_  ( A  X.  A ) )
20 endomtr 7388 . . . . 5  |-  ( ( A  ~~  ( A  X.  { (/) } )  /\  ( A  X.  { (/) } )  ~<_  ( A  X.  A ) )  ->  A  ~<_  ( A  X.  A ) )
215, 19, 20syl2anc 661 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  X.  A ) )
22 canth2g 7486 . . . . . . . . . 10  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
2322adantr 465 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<  ~P A
)
24 sdomdom 7358 . . . . . . . . 9  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
2523, 24syl 16 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ~P A )
26 xpdom1g 7429 . . . . . . . 8  |-  ( ( A  e. GCH  /\  A  ~<_  ~P A )  ->  ( A  X.  A )  ~<_  ( ~P A  X.  A
) )
2725, 26syldan 470 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ( ~P A  X.  A ) )
28 pwexg 4497 . . . . . . . . 9  |-  ( A  e. GCH  ->  ~P A  e. 
_V )
2928adantr 465 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P A  e.  _V )
30 xpdom2g 7428 . . . . . . . 8  |-  ( ( ~P A  e.  _V  /\  A  ~<_  ~P A )  -> 
( ~P A  X.  A )  ~<_  ( ~P A  X.  ~P A
) )
3129, 25, 30syl2anc 661 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  A )  ~<_  ( ~P A  X.  ~P A
) )
32 domtr 7383 . . . . . . 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 661 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ( ~P A  X.  ~P A ) )
34 simpl 457 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  e. GCH )
35 pwcdaen 8375 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A ) )
3634, 35syldan 470 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A
) )
3736ensymd 7381 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  ~P A )  ~~  ~P ( A  +c  A
) )
38 gchcdaidm 8856 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
39 pwen 7505 . . . . . . . 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 7382 . . . . . . 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 661 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  ~P A )  ~~  ~P A )
43 domentr 7389 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( ~P A  X.  ~P A )  /\  ( ~P A  X.  ~P A )  ~~  ~P A )  ->  ( A  X.  A )  ~<_  ~P A )
4433, 42, 43syl2anc 661 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ~P A )
45 gchinf 8845 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  om  ~<_  A )
46 pwxpndom 8854 . . . . . . 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 7379 . . . . . . 7  |-  ( ( A  X.  A ) 
~~  ~P A  ->  ~P A  ~~  ( A  X.  A ) )
49 endom 7357 . . . . . . 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 121 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ( A  X.  A )  ~~  ~P A )
52 brsdom 7353 . . . . 5  |-  ( ( A  X.  A ) 
~<  ~P A  <->  ( ( A  X.  A )  ~<_  ~P A  /\  -.  ( A  X.  A )  ~~  ~P A ) )
5344, 51, 52sylanbrc 664 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<  ~P A )
5421, 53jca 532 . . 3  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  ~<_  ( A  X.  A )  /\  ( A  X.  A
)  ~<  ~P A ) )
55 gchen1 8813 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  ( A  X.  A )  /\  ( A  X.  A )  ~<  ~P A
) )  ->  A  ~~  ( A  X.  A
) )
5654, 55mpdan 668 . 2  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  X.  A ) )
5756ensymd 7381 1  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~~  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2993   (/)c0 3658   ~Pcpw 3881   {csn 3898   class class class wbr 4313    X. cxp 4859  (class class class)co 6112   omcom 6497   1oc1o 6934    ~~ cen 7328    ~<_ cdom 7329    ~< csdm 7330   Fincfn 7331    +c ccda 8357  GCHcgch 8808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-seqom 6924  df-1o 6941  df-2o 6942  df-oadd 6945  df-omul 6946  df-oexp 6947  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-oi 7745  df-har 7794  df-cnf 7889  df-card 8130  df-cda 8358  df-fin4 8477  df-gch 8809
This theorem is referenced by:  gchhar  8867
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