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Theorem gchcdaidm 8258
Description: An infinite GCH-set is idempotent under cardinal sum. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
gchcdaidm  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )

Proof of Theorem gchcdaidm
StepHypRef Expression
1 simpl 445 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  e. GCH )
2 cdadom3 7782 . . . . 5  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  A  ~<_  ( A  +c  A ) )
31, 1, 2syl2anc 645 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  +c  A ) )
4 canth2g 6983 . . . . . . . . 9  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
54adantr 453 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<  ~P A
)
6 sdomdom 6857 . . . . . . . 8  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
75, 6syl 17 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ~P A )
8 cdadom1 7780 . . . . . . . 8  |-  ( A  ~<_  ~P A  ->  ( A  +c  A )  ~<_  ( ~P A  +c  A
) )
9 cdadom2 7781 . . . . . . . 8  |-  ( A  ~<_  ~P A  ->  ( ~P A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )
10 domtr 6882 . . . . . . . 8  |-  ( ( ( A  +c  A
)  ~<_  ( ~P A  +c  A )  /\  ( ~P A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )  ->  ( A  +c  A )  ~<_  ( ~P A  +c  ~P A
) )
118, 9, 10syl2anc 645 . . . . . . 7  |-  ( A  ~<_  ~P A  ->  ( A  +c  A )  ~<_  ( ~P A  +c  ~P A ) )
127, 11syl 17 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )
13 pwcda1 7788 . . . . . . . 8  |-  ( A  e. GCH  ->  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
1413adantr 453 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
15 gchcda1 8246 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  1o )  ~~  A )
16 pwen 7002 . . . . . . . 8  |-  ( ( A  +c  1o ) 
~~  A  ->  ~P ( A  +c  1o )  ~~  ~P A )
1715, 16syl 17 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  1o )  ~~  ~P A
)
18 entr 6881 . . . . . . 7  |-  ( ( ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o )  /\  ~P ( A  +c  1o )  ~~  ~P A )  ->  ( ~P A  +c  ~P A
)  ~~  ~P A
)
1914, 17, 18syl2anc 645 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P A )
20 domentr 6888 . . . . . 6  |-  ( ( ( A  +c  A
)  ~<_  ( ~P A  +c  ~P A )  /\  ( ~P A  +c  ~P A )  ~~  ~P A )  ->  ( A  +c  A )  ~<_  ~P A )
2112, 19, 20syl2anc 645 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<_  ~P A )
22 gchinf 8247 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  om  ~<_  A )
23 pwcdandom 8257 . . . . . . 7  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  A
) )
2422, 23syl 17 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ~P A  ~<_  ( A  +c  A ) )
25 ensym 6878 . . . . . . 7  |-  ( ( A  +c  A ) 
~~  ~P A  ->  ~P A  ~~  ( A  +c  A ) )
26 endom 6856 . . . . . . 7  |-  ( ~P A  ~~  ( A  +c  A )  ->  ~P A  ~<_  ( A  +c  A ) )
2725, 26syl 17 . . . . . 6  |-  ( ( A  +c  A ) 
~~  ~P A  ->  ~P A  ~<_  ( A  +c  A ) )
2824, 27nsyl 115 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ( A  +c  A )  ~~  ~P A )
29 brsdom 6852 . . . . 5  |-  ( ( A  +c  A ) 
~<  ~P A  <->  ( ( A  +c  A )  ~<_  ~P A  /\  -.  ( A  +c  A )  ~~  ~P A ) )
3021, 28, 29sylanbrc 648 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<  ~P A )
313, 30jca 520 . . 3  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  ~<_  ( A  +c  A )  /\  ( A  +c  A
)  ~<  ~P A ) )
32 gchen1 8215 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  ( A  +c  A )  /\  ( A  +c  A )  ~<  ~P A
) )  ->  A  ~~  ( A  +c  A
) )
3331, 32mpdan 652 . 2  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  +c  A ) )
34 ensym 6878 . 2  |-  ( A 
~~  ( A  +c  A )  ->  ( A  +c  A )  ~~  A )
3533, 34syl 17 1  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    e. wcel 1621   ~Pcpw 3599   class class class wbr 3997   omcom 4628  (class class class)co 5792   1oc1o 6440    ~~ cen 6828    ~<_ cdom 6829    ~< csdm 6830   Fincfn 6831    +c ccda 7761  GCHcgch 8210
This theorem is referenced by:  gchxpidm  8259  gchhar  8261  gchpwdom  8264
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-seqom 6428  df-1o 6447  df-2o 6448  df-oadd 6451  df-omul 6452  df-oexp 6453  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-oi 7193  df-har 7240  df-cnf 7331  df-card 7540  df-cda 7762  df-fin4 7881  df-gch 8211
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