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Theorem gchxpidm 9036
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 4569 . . . . . . . 8  |-  (/)  e.  _V
21a1i 11 . . . . . . 7  |-  ( -.  A  e.  Fin  ->  (/)  e.  _V )
3 xpsneng 7595 . . . . . . 7  |-  ( ( A  e. GCH  /\  (/)  e.  _V )  ->  ( A  X.  { (/) } )  ~~  A )
42, 3sylan2 472 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  { (/)
} )  ~~  A
)
54ensymd 7559 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  X.  { (/) } ) )
6 df1o2 7134 . . . . . . 7  |-  1o  =  { (/) }
7 id 22 . . . . . . . . . . . 12  |-  ( A  =  (/)  ->  A  =  (/) )
8 0fin 7740 . . . . . . . . . . . 12  |-  (/)  e.  Fin
97, 8syl6eqel 2550 . . . . . . . . . . 11  |-  ( A  =  (/)  ->  A  e. 
Fin )
109necon3bi 2683 . . . . . . . . . 10  |-  ( -.  A  e.  Fin  ->  A  =/=  (/) )
1110adantl 464 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  =/=  (/) )
12 0sdomg 7639 . . . . . . . . . 10  |-  ( A  e. GCH  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
1312adantr 463 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
1411, 13mpbird 232 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  -> 
(/)  ~<  A )
15 0sdom1dom 7710 . . . . . . . 8  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
1614, 15sylib 196 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  1o  ~<_  A )
176, 16syl5eqbrr 4473 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  { (/) }  ~<_  A )
18 xpdom2g 7606 . . . . . 6  |-  ( ( A  e. GCH  /\  { (/)
}  ~<_  A )  -> 
( A  X.  { (/)
} )  ~<_  ( A  X.  A ) )
1917, 18syldan 468 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  { (/)
} )  ~<_  ( A  X.  A ) )
20 endomtr 7566 . . . . 5  |-  ( ( A  ~~  ( A  X.  { (/) } )  /\  ( A  X.  { (/) } )  ~<_  ( A  X.  A ) )  ->  A  ~<_  ( A  X.  A ) )
215, 19, 20syl2anc 659 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  X.  A ) )
22 canth2g 7664 . . . . . . . . . 10  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
2322adantr 463 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<  ~P A
)
24 sdomdom 7536 . . . . . . . . 9  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
2523, 24syl 16 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ~P A )
26 xpdom1g 7607 . . . . . . . 8  |-  ( ( A  e. GCH  /\  A  ~<_  ~P A )  ->  ( A  X.  A )  ~<_  ( ~P A  X.  A
) )
2725, 26syldan 468 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ( ~P A  X.  A ) )
28 pwexg 4621 . . . . . . . . 9  |-  ( A  e. GCH  ->  ~P A  e. 
_V )
2928adantr 463 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P A  e.  _V )
30 xpdom2g 7606 . . . . . . . 8  |-  ( ( ~P A  e.  _V  /\  A  ~<_  ~P A )  -> 
( ~P A  X.  A )  ~<_  ( ~P A  X.  ~P A
) )
3129, 25, 30syl2anc 659 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  A )  ~<_  ( ~P A  X.  ~P A
) )
32 domtr 7561 . . . . . . 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 659 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ( ~P A  X.  ~P A ) )
34 simpl 455 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  e. GCH )
35 pwcdaen 8556 . . . . . . . . 9  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A ) )
3634, 35syldan 468 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  A )  ~~  ( ~P A  X.  ~P A
) )
3736ensymd 7559 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  ~P A )  ~~  ~P ( A  +c  A
) )
38 gchcdaidm 9035 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
39 pwen 7683 . . . . . . . 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 7560 . . . . . . 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 659 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  X.  ~P A )  ~~  ~P A )
43 domentr 7567 . . . . . 6  |-  ( ( ( A  X.  A
)  ~<_  ( ~P A  X.  ~P A )  /\  ( ~P A  X.  ~P A )  ~~  ~P A )  ->  ( A  X.  A )  ~<_  ~P A )
4433, 42, 43syl2anc 659 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<_  ~P A )
45 gchinf 9024 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  om  ~<_  A )
46 pwxpndom 9033 . . . . . . 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 7557 . . . . . . 7  |-  ( ( A  X.  A ) 
~~  ~P A  ->  ~P A  ~~  ( A  X.  A ) )
49 endom 7535 . . . . . . 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 7531 . . . . 5  |-  ( ( A  X.  A ) 
~<  ~P A  <->  ( ( A  X.  A )  ~<_  ~P A  /\  -.  ( A  X.  A )  ~~  ~P A ) )
5344, 51, 52sylanbrc 662 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  X.  A
)  ~<  ~P A )
5421, 53jca 530 . . 3  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  ~<_  ( A  X.  A )  /\  ( A  X.  A
)  ~<  ~P A ) )
55 gchen1 8992 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  ( A  X.  A )  /\  ( A  X.  A )  ~<  ~P A
) )  ->  A  ~~  ( A  X.  A
) )
5654, 55mpdan 666 . 2  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  X.  A ) )
5756ensymd 7559 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106   (/)c0 3783   ~Pcpw 3999   {csn 4016   class class class wbr 4439    X. cxp 4986  (class class class)co 6270   omcom 6673   1oc1o 7115    ~~ cen 7506    ~<_ cdom 7507    ~< csdm 7508   Fincfn 7509    +c ccda 8538  GCHcgch 8987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-oexp 7128  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-har 7976  df-cnf 8070  df-card 8311  df-cda 8539  df-fin4 8658  df-gch 8988
This theorem is referenced by:  gchhar  9046
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