Theorem canthp1lem1 9474

Description: Lemma for canthp1 9476. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
canthp1lem1	|- ( 1o ~< A -> ( A +c 2o ) ~<_ ~P A )

Proof of Theorem canthp1lem1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1sdom2 8159	. . 3 |- 1o ~< 2o
2		cdaxpdom 9011	. . 3 |- ( ( 1o ~< A /\ 1o ~< 2o ) -> ( A +c 2o ) ~<_ ( A X. 2o ) )
3	1, 2	mpan2 707	. 2 |- ( 1o ~< A -> ( A +c 2o ) ~<_ ( A X. 2o ) )
4		sdom0 8092	. . . . . 6 |- -. 1o ~< (/)
5		breq2 4657	. . . . . 6 |- ( A = (/) -> ( 1o ~< A <-> 1o ~< (/) ) )
6	4, 5	mtbiri 317	. . . . 5 |- ( A = (/) -> -. 1o ~< A )
7	6	con2i 134	. . . 4 |- ( 1o ~< A -> -. A = (/) )
8		neq0 3930	. . . 4 |- ( -. A = (/) <-> E. x x e. A )
9	7, 8	sylib 208	. . 3 |- ( 1o ~< A -> E. x x e. A )
10		relsdom 7962	. . . . . . . . . 10 |- Rel ~<
11	10	brrelex2i 5159	. . . . . . . . 9 |- ( 1o ~< A -> A e. _V )
12	11	adantr 481	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> A e. _V )
13		enrefg 7987	. . . . . . . 8 |- ( A e. _V -> A ~~ A )
14	12, 13	syl 17	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> A ~~ A )
15		df2o2 7574	. . . . . . . . 9 |- 2o = { (/) , { (/) } }
16		pwpw0 4344	. . . . . . . . 9 |- ~P { (/) } = { (/) , { (/) } }
17	15, 16	eqtr4i 2647	. . . . . . . 8 |- 2o = ~P { (/) }
18		0ex 4790	. . . . . . . . . 10 |- (/) e. _V
19		vex 3203	. . . . . . . . . 10 |- x e. _V
20		en2sn 8037	. . . . . . . . . 10 |- ( ( (/) e. _V /\ x e. _V ) -> { (/) } ~~ { x } )
21	18, 19, 20	mp2an 708	. . . . . . . . 9 |- { (/) } ~~ { x }
22		pwen 8133	. . . . . . . . 9 |- ( { (/) } ~~ { x } -> ~P { (/) } ~~ ~P { x } )
23	21, 22	ax-mp 5	. . . . . . . 8 |- ~P { (/) } ~~ ~P { x }
24	17, 23	eqbrtri 4674	. . . . . . 7 |- 2o ~~ ~P { x }
25		xpen 8123	. . . . . . 7 |- ( ( A ~~ A /\ 2o ~~ ~P { x } ) -> ( A X. 2o ) ~~ ( A X. ~P { x } ) )
26	14, 24, 25	sylancl 694	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~~ ( A X. ~P { x } ) )
27		snex 4908	. . . . . . . 8 |- { x } e. _V
28	27	pwex 4848	. . . . . . 7 |- ~P { x } e. _V
29		uncom 3757	. . . . . . . . 9 |- ( ( A \ { x } ) u. { x } ) = ( { x } u. ( A \ { x } ) )
30		simpr 477	. . . . . . . . . . 11 |- ( ( 1o ~< A /\ x e. A ) -> x e. A )
31	30	snssd 4340	. . . . . . . . . 10 |- ( ( 1o ~< A /\ x e. A ) -> { x } C_ A )
32		undif 4049	. . . . . . . . . 10 |- ( { x } C_ A <-> ( { x } u. ( A \ { x } ) ) = A )
33	31, 32	sylib 208	. . . . . . . . 9 |- ( ( 1o ~< A /\ x e. A ) -> ( { x } u. ( A \ { x } ) ) = A )
34	29, 33	syl5eq 2668	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) u. { x } ) = A )
35		difexg 4808	. . . . . . . . . 10 |- ( A e. _V -> ( A \ { x } ) e. _V )
36	12, 35	syl 17	. . . . . . . . 9 |- ( ( 1o ~< A /\ x e. A ) -> ( A \ { x } ) e. _V )
37		canth2g 8114	. . . . . . . . 9 |- ( ( A \ { x } ) e. _V -> ( A \ { x } ) ~< ~P ( A \ { x } ) )
38		domunsn 8110	. . . . . . . . 9 |- ( ( A \ { x } ) ~< ~P ( A \ { x } ) -> ( ( A \ { x } ) u. { x } ) ~<_ ~P ( A \ { x } ) )
39	36, 37, 38	3syl 18	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) u. { x } ) ~<_ ~P ( A \ { x } ) )
40	34, 39	eqbrtrrd 4677	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> A ~<_ ~P ( A \ { x } ) )
41		xpdom1g 8057	. . . . . . 7 |- ( ( ~P { x } e. _V /\ A ~<_ ~P ( A \ { x } ) ) -> ( A X. ~P { x } ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
42	28, 40, 41	sylancr 695	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. ~P { x } ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
43		endomtr 8014	. . . . . 6 |- ( ( ( A X. 2o ) ~~ ( A X. ~P { x } ) /\ ( A X. ~P { x } ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) ) -> ( A X. 2o ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
44	26, 42, 43	syl2anc 693	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
45		pwcdaen 9007	. . . . . . 7 |- ( ( ( A \ { x } ) e. _V /\ { x } e. _V ) -> ~P ( ( A \ { x } ) +c { x } ) ~~ ( ~P ( A \ { x } ) X. ~P { x } ) )
46	36, 27, 45	sylancl 694	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ~P ( ( A \ { x } ) +c { x } ) ~~ ( ~P ( A \ { x } ) X. ~P { x } ) )
47	46	ensymd 8007	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( ~P ( A \ { x } ) X. ~P { x } ) ~~ ~P ( ( A \ { x } ) +c { x } ) )
48		domentr 8015	. . . . 5 |- ( ( ( A X. 2o ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) /\ ( ~P ( A \ { x } ) X. ~P { x } ) ~~ ~P ( ( A \ { x } ) +c { x } ) ) -> ( A X. 2o ) ~<_ ~P ( ( A \ { x } ) +c { x } ) )
49	44, 47, 48	syl2anc 693	. . . 4 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~<_ ~P ( ( A \ { x } ) +c { x } ) )
50	27	a1i 11	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> { x } e. _V )
51		incom 3805	. . . . . . . . 9 |- ( ( A \ { x } ) i^i { x } ) = ( { x } i^i ( A \ { x } ) )
52		disjdif 4040	. . . . . . . . 9 |- ( { x } i^i ( A \ { x } ) ) = (/)
53	51, 52	eqtri 2644	. . . . . . . 8 |- ( ( A \ { x } ) i^i { x } ) = (/)
54	53	a1i 11	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) i^i { x } ) = (/) )
55		cdaun 8994	. . . . . . 7 |- ( ( ( A \ { x } ) e. _V /\ { x } e. _V /\ ( ( A \ { x } ) i^i { x } ) = (/) ) -> ( ( A \ { x } ) +c { x } ) ~~ ( ( A \ { x } ) u. { x } ) )
56	36, 50, 54, 55	syl3anc 1326	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) +c { x } ) ~~ ( ( A \ { x } ) u. { x } ) )
57	56, 34	breqtrd 4679	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) +c { x } ) ~~ A )
58		pwen 8133	. . . . 5 |- ( ( ( A \ { x } ) +c { x } ) ~~ A -> ~P ( ( A \ { x } ) +c { x } ) ~~ ~P A )
59	57, 58	syl 17	. . . 4 |- ( ( 1o ~< A /\ x e. A ) -> ~P ( ( A \ { x } ) +c { x } ) ~~ ~P A )
60		domentr 8015	. . . 4 |- ( ( ( A X. 2o ) ~<_ ~P ( ( A \ { x } ) +c { x } ) /\ ~P ( ( A \ { x } ) +c { x } ) ~~ ~P A ) -> ( A X. 2o ) ~<_ ~P A )
61	49, 59, 60	syl2anc 693	. . 3 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~<_ ~P A )
62	9, 61	exlimddv 1863	. 2 |- ( 1o ~< A -> ( A X. 2o ) ~<_ ~P A )
63		domtr 8009	. 2 |- ( ( ( A +c 2o ) ~<_ ( A X. 2o ) /\ ( A X. 2o ) ~<_ ~P A ) -> ( A +c 2o ) ~<_ ~P A )
64	3, 62, 63	syl2anc 693	1 |- ( 1o ~< A -> ( A +c 2o ) ~<_ ~P A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   class class class wbr 4653   X. cxp 5112  (class class class)co 6650   1oc1o 7553   2oc2o 7554   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   +c ccda 8989

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-cda 8990

This theorem is referenced by:  canthp1lem2  9475  canthp1  9476