Theorem canthp1lem1 9026

Description: Lemma for canthp1 9028. (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 7715	. . 3 |- 1o ~< 2o
2		cdaxpdom 8565	. . 3 |- ( ( 1o ~< A /\ 1o ~< 2o ) -> ( A +c 2o ) ~<_ ( A X. 2o ) )
3	1, 2	mpan2 671	. 2 |- ( 1o ~< A -> ( A +c 2o ) ~<_ ( A X. 2o ) )
4		sdom0 7646	. . . . . 6 |- -. 1o ~< (/)
5		breq2 4451	. . . . . 6 |- ( A = (/) -> ( 1o ~< A <-> 1o ~< (/) ) )
6	4, 5	mtbiri 303	. . . . 5 |- ( A = (/) -> -. 1o ~< A )
7	6	con2i 120	. . . 4 |- ( 1o ~< A -> -. A = (/) )
8		neq0 3795	. . . 4 |- ( -. A = (/) <-> E. x x e. A )
9	7, 8	sylib 196	. . 3 |- ( 1o ~< A -> E. x x e. A )
10		relsdom 7520	. . . . . . . . . 10 |- Rel ~<
11	10	brrelex2i 5040	. . . . . . . . 9 |- ( 1o ~< A -> A e. _V )
12	11	adantr 465	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> A e. _V )
13		enrefg 7544	. . . . . . . 8 |- ( A e. _V -> A ~~ A )
14	12, 13	syl 16	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> A ~~ A )
15		df2o2 7141	. . . . . . . . 9 |- 2o = { (/) , { (/) } }
16		pwpw0 4175	. . . . . . . . 9 |- ~P { (/) } = { (/) , { (/) } }
17	15, 16	eqtr4i 2499	. . . . . . . 8 |- 2o = ~P { (/) }
18		0ex 4577	. . . . . . . . . 10 |- (/) e. _V
19		vex 3116	. . . . . . . . . 10 |- x e. _V
20		en2sn 7592	. . . . . . . . . 10 |- ( ( (/) e. _V /\ x e. _V ) -> { (/) } ~~ { x } )
21	18, 19, 20	mp2an 672	. . . . . . . . 9 |- { (/) } ~~ { x }
22		pwen 7687	. . . . . . . . 9 |- ( { (/) } ~~ { x } -> ~P { (/) } ~~ ~P { x } )
23	21, 22	ax-mp 5	. . . . . . . 8 |- ~P { (/) } ~~ ~P { x }
24	17, 23	eqbrtri 4466	. . . . . . 7 |- 2o ~~ ~P { x }
25		xpen 7677	. . . . . . 7 |- ( ( A ~~ A /\ 2o ~~ ~P { x } ) -> ( A X. 2o ) ~~ ( A X. ~P { x } ) )
26	14, 24, 25	sylancl 662	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~~ ( A X. ~P { x } ) )
27		snex 4688	. . . . . . . 8 |- { x } e. _V
28	27	pwex 4630	. . . . . . 7 |- ~P { x } e. _V
29		uncom 3648	. . . . . . . . 9 |- ( ( A \ { x } ) u. { x } ) = ( { x } u. ( A \ { x } ) )
30		simpr 461	. . . . . . . . . . 11 |- ( ( 1o ~< A /\ x e. A ) -> x e. A )
31	30	snssd 4172	. . . . . . . . . 10 |- ( ( 1o ~< A /\ x e. A ) -> { x } C_ A )
32		undif 3907	. . . . . . . . . 10 |- ( { x } C_ A <-> ( { x } u. ( A \ { x } ) ) = A )
33	31, 32	sylib 196	. . . . . . . . 9 |- ( ( 1o ~< A /\ x e. A ) -> ( { x } u. ( A \ { x } ) ) = A )
34	29, 33	syl5eq 2520	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) u. { x } ) = A )
35		difexg 4595	. . . . . . . . . 10 |- ( A e. _V -> ( A \ { x } ) e. _V )
36	12, 35	syl 16	. . . . . . . . 9 |- ( ( 1o ~< A /\ x e. A ) -> ( A \ { x } ) e. _V )
37		canth2g 7668	. . . . . . . . 9 |- ( ( A \ { x } ) e. _V -> ( A \ { x } ) ~< ~P ( A \ { x } ) )
38		domunsn 7664	. . . . . . . . 9 |- ( ( A \ { x } ) ~< ~P ( A \ { x } ) -> ( ( A \ { x } ) u. { x } ) ~<_ ~P ( A \ { x } ) )
39	36, 37, 38	3syl 20	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) u. { x } ) ~<_ ~P ( A \ { x } ) )
40	34, 39	eqbrtrrd 4469	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> A ~<_ ~P ( A \ { x } ) )
41		xpdom1g 7611	. . . . . . 7 |- ( ( ~P { x } e. _V /\ A ~<_ ~P ( A \ { x } ) ) -> ( A X. ~P { x } ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
42	28, 40, 41	sylancr 663	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. ~P { x } ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
43		endomtr 7570	. . . . . 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 661	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
45		pwcdaen 8561	. . . . . . 7 |- ( ( ( A \ { x } ) e. _V /\ { x } e. _V ) -> ~P ( ( A \ { x } ) +c { x } ) ~~ ( ~P ( A \ { x } ) X. ~P { x } ) )
46	36, 27, 45	sylancl 662	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ~P ( ( A \ { x } ) +c { x } ) ~~ ( ~P ( A \ { x } ) X. ~P { x } ) )
47	46	ensymd 7563	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( ~P ( A \ { x } ) X. ~P { x } ) ~~ ~P ( ( A \ { x } ) +c { x } ) )
48		domentr 7571	. . . . 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 661	. . . 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 3691	. . . . . . . . 9 |- ( ( A \ { x } ) i^i { x } ) = ( { x } i^i ( A \ { x } ) )
52		disjdif 3899	. . . . . . . . 9 |- ( { x } i^i ( A \ { x } ) ) = (/)
53	51, 52	eqtri 2496	. . . . . . . 8 |- ( ( A \ { x } ) i^i { x } ) = (/)
54	53	a1i 11	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) i^i { x } ) = (/) )
55		cdaun 8548	. . . . . . 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 1228	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) +c { x } ) ~~ ( ( A \ { x } ) u. { x } ) )
57	56, 34	breqtrd 4471	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) +c { x } ) ~~ A )
58		pwen 7687	. . . . 5 |- ( ( ( A \ { x } ) +c { x } ) ~~ A -> ~P ( ( A \ { x } ) +c { x } ) ~~ ~P A )
59	57, 58	syl 16	. . . 4 |- ( ( 1o ~< A /\ x e. A ) -> ~P ( ( A \ { x } ) +c { x } ) ~~ ~P A )
60		domentr 7571	. . . 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 661	. . 3 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~<_ ~P A )
62	9, 61	exlimddv 1702	. 2 |- ( 1o ~< A -> ( A X. 2o ) ~<_ ~P A )
63		domtr 7565	. 2 |- ( ( ( A +c 2o ) ~<_ ( A X. 2o ) /\ ( A X. 2o ) ~<_ ~P A ) -> ( A +c 2o ) ~<_ ~P A )
64	3, 62, 63	syl2anc 661	1 |- ( 1o ~< A -> ( A +c 2o ) ~<_ ~P A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   _Vcvv 3113   \ cdif 3473   u. cun 3474   i^i cin 3475   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   {cpr 4029   class class class wbr 4447   X. cxp 4997  (class class class)co 6282   1oc1o 7120   2oc2o 7121   ~~ cen 7510   ~<_ cdom 7511   ~< csdm 7512   +c ccda 8543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-1o 7127  df-2o 7128  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-cda 8544

This theorem is referenced by:  canthp1lem2  9027  canthp1  9028