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Theorem gchcdaidm 8290
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 443 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  e. GCH )
2 cdadom3 7814 . . . . 5  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  A  ~<_  ( A  +c  A ) )
31, 1, 2syl2anc 642 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  +c  A ) )
4 canth2g 7015 . . . . . . . . 9  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
54adantr 451 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<  ~P A
)
6 sdomdom 6889 . . . . . . . 8  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
75, 6syl 15 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ~P A )
8 cdadom1 7812 . . . . . . . 8  |-  ( A  ~<_  ~P A  ->  ( A  +c  A )  ~<_  ( ~P A  +c  A
) )
9 cdadom2 7813 . . . . . . . 8  |-  ( A  ~<_  ~P A  ->  ( ~P A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )
10 domtr 6914 . . . . . . . 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 642 . . . . . . 7  |-  ( A  ~<_  ~P A  ->  ( A  +c  A )  ~<_  ( ~P A  +c  ~P A ) )
127, 11syl 15 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )
13 pwcda1 7820 . . . . . . . 8  |-  ( A  e. GCH  ->  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
1413adantr 451 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
15 gchcda1 8278 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  1o )  ~~  A )
16 pwen 7034 . . . . . . . 8  |-  ( ( A  +c  1o ) 
~~  A  ->  ~P ( A  +c  1o )  ~~  ~P A )
1715, 16syl 15 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  1o )  ~~  ~P A
)
18 entr 6913 . . . . . . 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 642 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P A )
20 domentr 6920 . . . . . 6  |-  ( ( ( A  +c  A
)  ~<_  ( ~P A  +c  ~P A )  /\  ( ~P A  +c  ~P A )  ~~  ~P A )  ->  ( A  +c  A )  ~<_  ~P A )
2112, 19, 20syl2anc 642 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<_  ~P A )
22 gchinf 8279 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  om  ~<_  A )
23 pwcdandom 8289 . . . . . . 7  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  A
) )
2422, 23syl 15 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ~P A  ~<_  ( A  +c  A ) )
25 ensym 6910 . . . . . . 7  |-  ( ( A  +c  A ) 
~~  ~P A  ->  ~P A  ~~  ( A  +c  A ) )
26 endom 6888 . . . . . . 7  |-  ( ~P A  ~~  ( A  +c  A )  ->  ~P A  ~<_  ( A  +c  A ) )
2725, 26syl 15 . . . . . 6  |-  ( ( A  +c  A ) 
~~  ~P A  ->  ~P A  ~<_  ( A  +c  A ) )
2824, 27nsyl 113 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ( A  +c  A )  ~~  ~P A )
29 brsdom 6884 . . . . 5  |-  ( ( A  +c  A ) 
~<  ~P A  <->  ( ( A  +c  A )  ~<_  ~P A  /\  -.  ( A  +c  A )  ~~  ~P A ) )
3021, 28, 29sylanbrc 645 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<  ~P A )
313, 30jca 518 . . 3  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  ~<_  ( A  +c  A )  /\  ( A  +c  A
)  ~<  ~P A ) )
32 gchen1 8247 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  ( A  +c  A )  /\  ( A  +c  A )  ~<  ~P A
) )  ->  A  ~~  ( A  +c  A
) )
3331, 32mpdan 649 . 2  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  +c  A ) )
34 ensym 6910 . 2  |-  ( A 
~~  ( A  +c  A )  ->  ( A  +c  A )  ~~  A )
3533, 34syl 15 1  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1684   ~Pcpw 3625   class class class wbr 4023   omcom 4656  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863    +c ccda 7793  GCHcgch 8242
This theorem is referenced by:  gchxpidm  8291  gchhar  8293  gchpwdom  8296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-cnf 7363  df-card 7572  df-cda 7794  df-fin4 7913  df-gch 8243
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