Theorem canthp1lem1 9019

Description: Lemma for canthp1 9021. (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 7711	. . 3 |- 1o ~< 2o
2		cdaxpdom 8560	. . 3 |- ( ( 1o ~< A /\ 1o ~< 2o ) -> ( A +c 2o ) ~<_ ( A X. 2o ) )
3	1, 2	mpan2 669	. 2 |- ( 1o ~< A -> ( A +c 2o ) ~<_ ( A X. 2o ) )
4		sdom0 7642	. . . . . 6 |- -. 1o ~< (/)
5		breq2 4443	. . . . . 6 |- ( A = (/) -> ( 1o ~< A <-> 1o ~< (/) ) )
6	4, 5	mtbiri 301	. . . . 5 |- ( A = (/) -> -. 1o ~< A )
7	6	con2i 120	. . . 4 |- ( 1o ~< A -> -. A = (/) )
8		neq0 3794	. . . 4 |- ( -. A = (/) <-> E. x x e. A )
9	7, 8	sylib 196	. . 3 |- ( 1o ~< A -> E. x x e. A )
10		relsdom 7516	. . . . . . . . . 10 |- Rel ~<
11	10	brrelex2i 5030	. . . . . . . . 9 |- ( 1o ~< A -> A e. _V )
12	11	adantr 463	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> A e. _V )
13		enrefg 7540	. . . . . . . 8 |- ( A e. _V -> A ~~ A )
14	12, 13	syl 16	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> A ~~ A )
15		df2o2 7136	. . . . . . . . 9 |- 2o = { (/) , { (/) } }
16		pwpw0 4164	. . . . . . . . 9 |- ~P { (/) } = { (/) , { (/) } }
17	15, 16	eqtr4i 2486	. . . . . . . 8 |- 2o = ~P { (/) }
18		0ex 4569	. . . . . . . . . 10 |- (/) e. _V
19		vex 3109	. . . . . . . . . 10 |- x e. _V
20		en2sn 7588	. . . . . . . . . 10 |- ( ( (/) e. _V /\ x e. _V ) -> { (/) } ~~ { x } )
21	18, 19, 20	mp2an 670	. . . . . . . . 9 |- { (/) } ~~ { x }
22		pwen 7683	. . . . . . . . 9 |- ( { (/) } ~~ { x } -> ~P { (/) } ~~ ~P { x } )
23	21, 22	ax-mp 5	. . . . . . . 8 |- ~P { (/) } ~~ ~P { x }
24	17, 23	eqbrtri 4458	. . . . . . 7 |- 2o ~~ ~P { x }
25		xpen 7673	. . . . . . 7 |- ( ( A ~~ A /\ 2o ~~ ~P { x } ) -> ( A X. 2o ) ~~ ( A X. ~P { x } ) )
26	14, 24, 25	sylancl 660	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~~ ( A X. ~P { x } ) )
27		snex 4678	. . . . . . . 8 |- { x } e. _V
28	27	pwex 4620	. . . . . . 7 |- ~P { x } e. _V
29		uncom 3634	. . . . . . . . 9 |- ( ( A \ { x } ) u. { x } ) = ( { x } u. ( A \ { x } ) )
30		simpr 459	. . . . . . . . . . 11 |- ( ( 1o ~< A /\ x e. A ) -> x e. A )
31	30	snssd 4161	. . . . . . . . . 10 |- ( ( 1o ~< A /\ x e. A ) -> { x } C_ A )
32		undif 3896	. . . . . . . . . 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 2507	. . . . . . . 8 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) u. { x } ) = A )
35		difexg 4585	. . . . . . . . . 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 7664	. . . . . . . . 9 |- ( ( A \ { x } ) e. _V -> ( A \ { x } ) ~< ~P ( A \ { x } ) )
38		domunsn 7660	. . . . . . . . 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 4461	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> A ~<_ ~P ( A \ { x } ) )
41		xpdom1g 7607	. . . . . . 7 |- ( ( ~P { x } e. _V /\ A ~<_ ~P ( A \ { x } ) ) -> ( A X. ~P { x } ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
42	28, 40, 41	sylancr 661	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. ~P { x } ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
43		endomtr 7566	. . . . . 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 659	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~<_ ( ~P ( A \ { x } ) X. ~P { x } ) )
45		pwcdaen 8556	. . . . . . 7 |- ( ( ( A \ { x } ) e. _V /\ { x } e. _V ) -> ~P ( ( A \ { x } ) +c { x } ) ~~ ( ~P ( A \ { x } ) X. ~P { x } ) )
46	36, 27, 45	sylancl 660	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ~P ( ( A \ { x } ) +c { x } ) ~~ ( ~P ( A \ { x } ) X. ~P { x } ) )
47	46	ensymd 7559	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( ~P ( A \ { x } ) X. ~P { x } ) ~~ ~P ( ( A \ { x } ) +c { x } ) )
48		domentr 7567	. . . . 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 659	. . . 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 3677	. . . . . . . . 9 |- ( ( A \ { x } ) i^i { x } ) = ( { x } i^i ( A \ { x } ) )
52		disjdif 3888	. . . . . . . . 9 |- ( { x } i^i ( A \ { x } ) ) = (/)
53	51, 52	eqtri 2483	. . . . . . . 8 |- ( ( A \ { x } ) i^i { x } ) = (/)
54	53	a1i 11	. . . . . . 7 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) i^i { x } ) = (/) )
55		cdaun 8543	. . . . . . 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 1226	. . . . . 6 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) +c { x } ) ~~ ( ( A \ { x } ) u. { x } ) )
57	56, 34	breqtrd 4463	. . . . 5 |- ( ( 1o ~< A /\ x e. A ) -> ( ( A \ { x } ) +c { x } ) ~~ A )
58		pwen 7683	. . . . 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 7567	. . . 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 659	. . 3 |- ( ( 1o ~< A /\ x e. A ) -> ( A X. 2o ) ~<_ ~P A )
62	9, 61	exlimddv 1731	. 2 |- ( 1o ~< A -> ( A X. 2o ) ~<_ ~P A )
63		domtr 7561	. 2 |- ( ( ( A +c 2o ) ~<_ ( A X. 2o ) /\ ( A X. 2o ) ~<_ ~P A ) -> ( A +c 2o ) ~<_ ~P A )
64	3, 62, 63	syl2anc 659	1 |- ( 1o ~< A -> ( A +c 2o ) ~<_ ~P A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   = wceq 1398   E.wex 1617   e. wcel 1823   _Vcvv 3106   \ cdif 3458   u. cun 3459   i^i cin 3460   C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   {cpr 4018   class class class wbr 4439   X. cxp 4986  (class class class)co 6270   1oc1o 7115   2oc2o 7116   ~~ cen 7506   ~<_ cdom 7507   ~< csdm 7508   +c ccda 8538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-1o 7122  df-2o 7123  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-cda 8539

This theorem is referenced by:  canthp1lem2  9020  canthp1  9021