Theorem canthp1lem1 8906

Description: Lemma for canthp1 8908. (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 7598	. . 3 |- 1o ~< 2o
2		cdaxpdom 8445	. . 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 7529	. . . . . 6 |- -. 1o ~< (/)
5		breq2 4380	. . . . . 6 |- ( A = (/) -> ( 1o ~< A <-> 1o ~< (/) ) )
6	4, 5	mtbiri 303	. . . . 5 |- ( A = (/) -> -. 1o ~< A )
7	6	con2i 120	. . . 4 |- ( 1o ~< A -> -. A = (/) )
8		neq0 3731	. . . 4 |- ( -. A = (/) <-> E. x x e. A )
9	7, 8	sylib 196	. . 3 |- ( 1o ~< A -> E. x x e. A )
10		relsdom 7403	. . . . . . . . . 10 |- Rel ~<
11	10	brrelex2i 4964	. . . . . . . . 9 |- ( 1o ~< A -> A e. _V )
12	11	adantr 465	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> A e. _V )
13		enrefg 7427	. . . . . . . 8 |- ( A e. _V -> A ~~ A )
14	12, 13	syl 16	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> A ~~ A )
15		df2o2 7020	. . . . . . . . 9 |- 2o = { (/) , { (/) } }
16		pwpw0 4105	. . . . . . . . 9 |- ~P { (/) } = { (/) , { (/) } }
17	15, 16	eqtr4i 2481	. . . . . . . 8 |- 2o = ~P { (/) }
18		0ex 4506	. . . . . . . . . 10 |- (/) e. _V
19		vex 3057	. . . . . . . . . 10 |- x e. _V
20		en2sn 7475	. . . . . . . . . 10 |- ( ( (/) e. _V /\ x e. _V ) -> { (/) } ~~ { x } )
21	18, 19, 20	mp2an 672	. . . . . . . . 9 |- { (/) } ~~ { x }
22		pwen 7570	. . . . . . . . 9 |- ( { (/) } ~~ { x } -> ~P { (/) } ~~ ~P { x } )
23	21, 22	ax-mp 5	. . . . . . . 8 |- ~P { (/) } ~~ ~P { x }
24	17, 23	eqbrtri 4395	. . . . . . 7 |- 2o ~~ ~P { x }
25		xpen 7560	. . . . . . 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 4617	. . . . . . . 8 |- { x } e. _V
28	27	pwex 4559	. . . . . . 7 |- ~P { x } e. _V
29		uncom 3584	. . . . . . . . 9 |- ( ( A \ { x } ) u. { x } ) = ( { x } u. ( A \ { x } ) )
30		simpr 461	. . . . . . . . . . 11 |- ( ( 1o ~< A /\ x e. A ) -> x e. A )
31	30	snssd 4102	. . . . . . . . . 10 |- ( ( 1o ~< A /\ x e. A ) -> { x } C_ A )
32		undif 3843	. . . . . . . . . 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 2502	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) u. { x } ) = A )
35		difexg 4524	. . . . . . . . . 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 7551	. . . . . . . . 9 |- ( ( A \ { x } ) e. _V -> ( A \ { x } ) ~< ~P ( A \ { x } ) )
38		domunsn 7547	. . . . . . . . 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 4398	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> A ~<_ ~P ( A \ { x } ) )
41		xpdom1g 7494	. . . . . . 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 7453	. . . . . 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 8441	. . . . . . 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 7446	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( ~P ( A \ { x } ) X. ~P { x } ) ~~ ~P ( ( A \ { x } ) +c { x } ) )
48		domentr 7454	. . . . 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 3627	. . . . . . . . 9 |- ( ( A \ { x } ) i^i { x } ) = ( { x } i^i ( A \ { x } ) )
52		disjdif 3835	. . . . . . . . 9 |- ( { x } i^i ( A \ { x } ) ) = (/)
53	51, 52	eqtri 2478	. . . . . . . 8 |- ( ( A \ { x } ) i^i { x } ) = (/)
54	53	a1i 11	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) i^i { x } ) = (/) )
55		cdaun 8428	. . . . . . 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 1219	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) +c { x } ) ~~ ( ( A \ { x } ) u. { x } ) )
57	56, 34	breqtrd 4400	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) +c { x } ) ~~ A )
58		pwen 7570	. . . . 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 7454	. . . 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 1693	. 2 |- ( 1o ~< A -> ( A X. 2o ) ~<_ ~P A )
63		domtr 7448	. 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 1370   E.wex 1587   e. wcel 1757   _Vcvv 3054   \ cdif 3409   u. cun 3410   i^i cin 3411   C_ wss 3412   (/)c0 3721   ~Pcpw 3944   {csn 3961   {cpr 3963   class class class wbr 4376   X. cxp 4922  (class class class)co 6176   1oc1o 6999   2oc2o 7000   ~~ cen 7393   ~<_ cdom 7394   ~< csdm 7395   +c ccda 8423

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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-1o 7006  df-2o 7007  df-er 7187  df-map 7302  df-en 7397  df-dom 7398  df-sdom 7399  df-cda 8424

This theorem is referenced by:  canthp1lem2  8907  canthp1  8908