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Theorem gchcdaidm 9490
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 473 . . . . 5 |- ( ( A e. GCH /\ -. A e. Fin ) -> A e. GCH )
2 cdadom3 9010 . . . . 5 |- ( ( A e. GCH /\ A e. GCH ) -> A ~<_ ( A +c A ) )
31, 1, 2syl2anc 693 . . . 4 |- ( ( A e. GCH /\ -. A e. Fin ) -> A ~<_ ( A +c A ) )
4 canth2g 8114 . . . . . . . . 9 |- ( A e. GCH -> A ~< ~P A )
54adantr 481 . . . . . . . 8 |- ( ( A e. GCH /\ -. A e. Fin ) -> A ~< ~P A )
6 sdomdom 7983 . . . . . . . 8 |- ( A ~< ~P A -> A ~<_ ~P A )
75, 6syl 17 . . . . . . 7 |- ( ( A e. GCH /\ -. A e. Fin ) -> A ~<_ ~P A )
8 cdadom1 9008 . . . . . . . 8 |- ( A ~<_ ~P A -> ( A +c A ) ~<_ ( ~P A +c A ) )
9 cdadom2 9009 . . . . . . . 8 |- ( A ~<_ ~P A -> ( ~P A +c A ) ~<_ ( ~P A +c ~P A ) )
10 domtr 8009 . . . . . . . 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 693 . . . . . . 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 9016 . . . . . . . 8 |- ( A e. GCH -> ( ~P A +c ~P A ) ~~ ~P ( A +c 1o ) )
1413adantr 481 . . . . . . 7 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( ~P A +c ~P A ) ~~ ~P ( A +c 1o ) )
15 gchcda1 9478 . . . . . . . 8 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A +c 1o ) ~~ A )
16 pwen 8133 . . . . . . . 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 8008 . . . . . . 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 693 . . . . . 6 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( ~P A +c ~P A ) ~~ ~P A )
20 domentr 8015 . . . . . 6 |- ( ( ( A +c A ) ~<_ ( ~P A +c ~P A ) /\ ( ~P A +c ~P A ) ~~ ~P A ) -> ( A +c A ) ~<_ ~P A )
2112, 19, 20syl2anc 693 . . . . 5 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A +c A ) ~<_ ~P A )
22 gchinf 9479 . . . . . . 7 |- ( ( A e. GCH /\ -. A e. Fin ) -> om ~<_ A )
23 pwcdandom 9489 . . . . . . 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 8005 . . . . . . 7 |- ( ( A +c A ) ~~ ~P A -> ~P A ~~ ( A +c A ) )
26 endom 7982 . . . . . . 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 135 . . . . 5 |- ( ( A e. GCH /\ -. A e. Fin ) -> -. ( A +c A ) ~~ ~P A )
29 brsdom 7978 . . . . 5 |- ( ( A +c A ) ~< ~P A <-> ( ( A +c A ) ~<_ ~P A /\ -. ( A +c A ) ~~ ~P A ) )
3021, 28, 29sylanbrc 698 . . . 4 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A +c A ) ~< ~P A )
313, 30jca 554 . . 3 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A ~<_ ( A +c A ) /\ ( A +c A ) ~< ~P A ) )
32 gchen1 9447 . . 3 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ ( A +c A ) /\ ( A +c A ) ~< ~P A ) ) -> A ~~ ( A +c A ) )
3331, 32mpdan 702 . 2 |- ( ( A e. GCH /\ -. A e. Fin ) -> A ~~ ( A +c A ) )
3433ensymd 8007 1 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A +c A ) ~~ A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   ~Pcpw 4158   class class class wbr 4653  (class class class)co 6650   omcom 7065   1oc1o 7553   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   +c ccda 8989  GCHcgch 9442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-har 8463  df-cnf 8559  df-card 8765  df-cda 8990  df-fin4 9109  df-gch 9443
This theorem is referenced by:  gchxpidm  9491  gchpwdom  9492  gchhar  9501
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