Theorem gchxpidm 9491

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

Step	Hyp	Ref	Expression
1		0ex 4790	. . . . . . . 8 |- (/) e. _V
2	1	a1i 11	. . . . . . 7 |- ( -. A e. Fin -> (/) e. _V )
3		xpsneng 8045	. . . . . . 7 |- ( ( A e. GCH /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
4	2, 3	sylan2 491	. . . . . 6 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. { (/) } ) ~~ A )
5	4	ensymd 8007	. . . . 5 |- ( ( A e. GCH /\ -. A e. Fin ) -> A ~~ ( A X. { (/) } ) )
6		df1o2 7572	. . . . . . 7 |- 1o = { (/) }
7		id 22	. . . . . . . . . . . 12 |- ( A = (/) -> A = (/) )
8		0fin 8188	. . . . . . . . . . . 12 |- (/) e. Fin
9	7, 8	syl6eqel 2709	. . . . . . . . . . 11 |- ( A = (/) -> A e. Fin )
10	9	necon3bi 2820	. . . . . . . . . 10 |- ( -. A e. Fin -> A =/= (/) )
11	10	adantl 482	. . . . . . . . 9 |- ( ( A e. GCH /\ -. A e. Fin ) -> A =/= (/) )
12		0sdomg 8089	. . . . . . . . . 10 |- ( A e. GCH -> ( (/) ~< A <-> A =/= (/) ) )
13	12	adantr 481	. . . . . . . . 9 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( (/) ~< A <-> A =/= (/) ) )
14	11, 13	mpbird 247	. . . . . . . 8 |- ( ( A e. GCH /\ -. A e. Fin ) -> (/) ~< A )
15		0sdom1dom 8158	. . . . . . . 8 |- ( (/) ~< A <-> 1o ~<_ A )
16	14, 15	sylib 208	. . . . . . 7 |- ( ( A e. GCH /\ -. A e. Fin ) -> 1o ~<_ A )
17	6, 16	syl5eqbrr 4689	. . . . . 6 |- ( ( A e. GCH /\ -. A e. Fin ) -> { (/) } ~<_ A )
18		xpdom2g 8056	. . . . . 6 |- ( ( A e. GCH /\ { (/) } ~<_ A ) -> ( A X. { (/) } ) ~<_ ( A X. A ) )
19	17, 18	syldan 487	. . . . 5 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. { (/) } ) ~<_ ( A X. A ) )
20		endomtr 8014	. . . . 5 |- ( ( A ~~ ( A X. { (/) } ) /\ ( A X. { (/) } ) ~<_ ( A X. A ) ) -> A ~<_ ( A X. A ) )
21	5, 19, 20	syl2anc 693	. . . 4 |- ( ( A e. GCH /\ -. A e. Fin ) -> A ~<_ ( A X. A ) )
22		canth2g 8114	. . . . . . . . . 10 |- ( A e. GCH -> A ~< ~P A )
23	22	adantr 481	. . . . . . . . 9 |- ( ( A e. GCH /\ -. A e. Fin ) -> A ~< ~P A )
24		sdomdom 7983	. . . . . . . . 9 |- ( A ~< ~P A -> A ~<_ ~P A )
25	23, 24	syl 17	. . . . . . . 8 |- ( ( A e. GCH /\ -. A e. Fin ) -> A ~<_ ~P A )
26		xpdom1g 8057	. . . . . . . 8 |- ( ( A e. GCH /\ A ~<_ ~P A ) -> ( A X. A ) ~<_ ( ~P A X. A ) )
27	25, 26	syldan 487	. . . . . . 7 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. A ) ~<_ ( ~P A X. A ) )
28		pwexg 4850	. . . . . . . . 9 |- ( A e. GCH -> ~P A e. _V )
29	28	adantr 481	. . . . . . . 8 |- ( ( A e. GCH /\ -. A e. Fin ) -> ~P A e. _V )
30		xpdom2g 8056	. . . . . . . 8 |- ( ( ~P A e. _V /\ A ~<_ ~P A ) -> ( ~P A X. A ) ~<_ ( ~P A X. ~P A ) )
31	29, 25, 30	syl2anc 693	. . . . . . 7 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( ~P A X. A ) ~<_ ( ~P A X. ~P A ) )
32		domtr 8009	. . . . . . 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 ) )
33	27, 31, 32	syl2anc 693	. . . . . 6 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. A ) ~<_ ( ~P A X. ~P A ) )
34		simpl 473	. . . . . . . . 9 |- ( ( A e. GCH /\ -. A e. Fin ) -> A e. GCH )
35		pwcdaen 9007	. . . . . . . . 9 |- ( ( A e. GCH /\ A e. GCH ) -> ~P ( A +c A ) ~~ ( ~P A X. ~P A ) )
36	34, 35	syldan 487	. . . . . . . 8 |- ( ( A e. GCH /\ -. A e. Fin ) -> ~P ( A +c A ) ~~ ( ~P A X. ~P A ) )
37	36	ensymd 8007	. . . . . . 7 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( ~P A X. ~P A ) ~~ ~P ( A +c A ) )
38		gchcdaidm 9490	. . . . . . . 8 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A +c A ) ~~ A )
39		pwen 8133	. . . . . . . 8 |- ( ( A +c A ) ~~ A -> ~P ( A +c A ) ~~ ~P A )
40	38, 39	syl 17	. . . . . . 7 |- ( ( A e. GCH /\ -. A e. Fin ) -> ~P ( A +c A ) ~~ ~P A )
41		entr 8008	. . . . . . 7 |- ( ( ( ~P A X. ~P A ) ~~ ~P ( A +c A ) /\ ~P ( A +c A ) ~~ ~P A ) -> ( ~P A X. ~P A ) ~~ ~P A )
42	37, 40, 41	syl2anc 693	. . . . . 6 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( ~P A X. ~P A ) ~~ ~P A )
43		domentr 8015	. . . . . 6 |- ( ( ( A X. A ) ~<_ ( ~P A X. ~P A ) /\ ( ~P A X. ~P A ) ~~ ~P A ) -> ( A X. A ) ~<_ ~P A )
44	33, 42, 43	syl2anc 693	. . . . 5 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. A ) ~<_ ~P A )
45		gchinf 9479	. . . . . . 7 |- ( ( A e. GCH /\ -. A e. Fin ) -> om ~<_ A )
46		pwxpndom 9488	. . . . . . 7 |- ( om ~<_ A -> -. ~P A ~<_ ( A X. A ) )
47	45, 46	syl 17	. . . . . 6 |- ( ( A e. GCH /\ -. A e. Fin ) -> -. ~P A ~<_ ( A X. A ) )
48		ensym 8005	. . . . . . 7 |- ( ( A X. A ) ~~ ~P A -> ~P A ~~ ( A X. A ) )
49		endom 7982	. . . . . . 7 |- ( ~P A ~~ ( A X. A ) -> ~P A ~<_ ( A X. A ) )
50	48, 49	syl 17	. . . . . 6 |- ( ( A X. A ) ~~ ~P A -> ~P A ~<_ ( A X. A ) )
51	47, 50	nsyl 135	. . . . 5 |- ( ( A e. GCH /\ -. A e. Fin ) -> -. ( A X. A ) ~~ ~P A )
52		brsdom 7978	. . . . 5 |- ( ( A X. A ) ~< ~P A <-> ( ( A X. A ) ~<_ ~P A /\ -. ( A X. A ) ~~ ~P A ) )
53	44, 51, 52	sylanbrc 698	. . . 4 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. A ) ~< ~P A )
54	21, 53	jca 554	. . 3 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A ~<_ ( A X. A ) /\ ( A X. A ) ~< ~P A ) )
55		gchen1 9447	. . 3 |- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ ( A X. A ) /\ ( A X. A ) ~< ~P A ) ) -> A ~~ ( A X. A ) )
56	54, 55	mpdan 702	. 2 |- ( ( A e. GCH /\ -. A e. Fin ) -> A ~~ ( A X. A ) )
57	56	ensymd 8007	1 |- ( ( A e. GCH /\ -. A e. Fin ) -> ( A X. A ) ~~ A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   X. cxp 5112  (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:  gchhar  9501