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Theorem gchcdaidm 8939
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 457 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  e. GCH )
2 cdadom3 8461 . . . . 5  |-  ( ( A  e. GCH  /\  A  e. GCH )  ->  A  ~<_  ( A  +c  A ) )
31, 1, 2syl2anc 661 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ( A  +c  A ) )
4 canth2g 7568 . . . . . . . . 9  |-  ( A  e. GCH  ->  A  ~<  ~P A
)
54adantr 465 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<  ~P A
)
6 sdomdom 7440 . . . . . . . 8  |-  ( A 
~<  ~P A  ->  A  ~<_  ~P A )
75, 6syl 16 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~<_  ~P A )
8 cdadom1 8459 . . . . . . . 8  |-  ( A  ~<_  ~P A  ->  ( A  +c  A )  ~<_  ( ~P A  +c  A
) )
9 cdadom2 8460 . . . . . . . 8  |-  ( A  ~<_  ~P A  ->  ( ~P A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )
10 domtr 7465 . . . . . . . 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 661 . . . . . . 7  |-  ( A  ~<_  ~P A  ->  ( A  +c  A )  ~<_  ( ~P A  +c  ~P A ) )
127, 11syl 16 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<_  ( ~P A  +c  ~P A ) )
13 pwcda1 8467 . . . . . . . 8  |-  ( A  e. GCH  ->  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
1413adantr 465 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P ( A  +c  1o ) )
15 gchcda1 8927 . . . . . . . 8  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  1o )  ~~  A )
16 pwen 7587 . . . . . . . 8  |-  ( ( A  +c  1o ) 
~~  A  ->  ~P ( A  +c  1o )  ~~  ~P A )
1715, 16syl 16 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ~P ( A  +c  1o )  ~~  ~P A
)
18 entr 7464 . . . . . . 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 661 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( ~P A  +c  ~P A )  ~~  ~P A )
20 domentr 7471 . . . . . 6  |-  ( ( ( A  +c  A
)  ~<_  ( ~P A  +c  ~P A )  /\  ( ~P A  +c  ~P A )  ~~  ~P A )  ->  ( A  +c  A )  ~<_  ~P A )
2112, 19, 20syl2anc 661 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<_  ~P A )
22 gchinf 8928 . . . . . . 7  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  om  ~<_  A )
23 pwcdandom 8938 . . . . . . 7  |-  ( om  ~<_  A  ->  -.  ~P A  ~<_  ( A  +c  A
) )
2422, 23syl 16 . . . . . 6  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ~P A  ~<_  ( A  +c  A ) )
25 ensym 7461 . . . . . . 7  |-  ( ( A  +c  A ) 
~~  ~P A  ->  ~P A  ~~  ( A  +c  A ) )
26 endom 7439 . . . . . . 7  |-  ( ~P A  ~~  ( A  +c  A )  ->  ~P A  ~<_  ( A  +c  A ) )
2725, 26syl 16 . . . . . 6  |-  ( ( A  +c  A ) 
~~  ~P A  ->  ~P A  ~<_  ( A  +c  A ) )
2824, 27nsyl 121 . . . . 5  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  -.  ( A  +c  A )  ~~  ~P A )
29 brsdom 7435 . . . . 5  |-  ( ( A  +c  A ) 
~<  ~P A  <->  ( ( A  +c  A )  ~<_  ~P A  /\  -.  ( A  +c  A )  ~~  ~P A ) )
3021, 28, 29sylanbrc 664 . . . 4  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~<  ~P A )
313, 30jca 532 . . 3  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  ~<_  ( A  +c  A )  /\  ( A  +c  A
)  ~<  ~P A ) )
32 gchen1 8896 . . 3  |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  ( A  +c  A )  /\  ( A  +c  A )  ~<  ~P A
) )  ->  A  ~~  ( A  +c  A
) )
3331, 32mpdan 668 . 2  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  A  ~~  ( A  +c  A ) )
3433ensymd 7463 1  |-  ( ( A  e. GCH  /\  -.  A  e.  Fin )  ->  ( A  +c  A
)  ~~  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1758   ~Pcpw 3961   class class class wbr 4393  (class class class)co 6193   omcom 6579   1oc1o 7016    ~~ cen 7410    ~<_ cdom 7411    ~< csdm 7412   Fincfn 7413    +c ccda 8440  GCHcgch 8891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-seqom 7006  df-1o 7023  df-2o 7024  df-oadd 7027  df-omul 7028  df-oexp 7029  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-oi 7828  df-har 7877  df-cnf 7972  df-card 8213  df-cda 8441  df-fin4 8560  df-gch 8892
This theorem is referenced by:  gchxpidm  8940  gchpwdom  8941  gchhar  8950
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