Theorem pwle2 14833

Description: An exercise related to N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)

Hypothesis

Ref	Expression
pwle2.t	|- T = U_ x e. N ( { x } X. 1o )

Assertion

Ref	Expression
pwle2	|- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> N C_ 2o )

Distinct variable group:   x, N

Allowed substitution hints:   T( x)   G( x)

Proof of Theorem pwle2

Step	Hyp	Ref	Expression
1		simplr 528	. . . . . . . . . . 11 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> G : T -1-1-> ~P 1o )
2		f1f 5423	. . . . . . . . . . 11 |- ( G : T -1-1-> ~P 1o -> G : T --> ~P 1o )
3	1, 2	syl 14	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> G : T --> ~P 1o )
4		nnon 4611	. . . . . . . . . . . . . 14 |- ( N e. om -> N e. On )
5	4	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> N e. On )
6		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> 2o e. N )
7		1lt2o 6445	. . . . . . . . . . . . . 14 |- 1o e. 2o
8	6, 7	jctil 312	. . . . . . . . . . . . 13 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( 1o e. 2o /\ 2o e. N ) )
9		ontr1 4391	. . . . . . . . . . . . 13 |- ( N e. On -> ( ( 1o e. 2o /\ 2o e. N ) -> 1o e. N ) )
10	5, 8, 9	sylc 62	. . . . . . . . . . . 12 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> 1o e. N )
11		0lt1o 6443	. . . . . . . . . . . 12 |- (/) e. 1o
12		opelxpi 4660	. . . . . . . . . . . 12 |- ( ( 1o e. N /\ (/) e. 1o ) -> <. 1o , (/) >. e. ( N X. 1o ) )
13	10, 11, 12	sylancl 413	. . . . . . . . . . 11 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> <. 1o , (/) >. e. ( N X. 1o ) )
14		pwle2.t	. . . . . . . . . . . 12 |- T = U_ x e. N ( { x } X. 1o )
15		iunxpconst 4688	. . . . . . . . . . . 12 |- U_ x e. N ( { x } X. 1o ) = ( N X. 1o )
16	14, 15	eqtri 2198	. . . . . . . . . . 11 |- T = ( N X. 1o )
17	13, 16	eleqtrrdi 2271	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> <. 1o , (/) >. e. T )
18	3, 17	ffvelcdmd 5654	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( G ` <. 1o , (/) >. ) e. ~P 1o )
19	18	elpwid 3588	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( G ` <. 1o , (/) >. ) C_ 1o )
20		df1o2 6432	. . . . . . . 8 |- 1o = { (/) }
21	19, 20	sseqtrdi 3205	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( G ` <. 1o , (/) >. ) C_ { (/) } )
22		pwtrufal 14832	. . . . . . 7 |- ( ( G ` <. 1o , (/) >. ) C_ { (/) } -> -. -. ( ( G ` <. 1o , (/) >. ) = (/) \/ ( G ` <. 1o , (/) >. ) = { (/) } ) )
23	21, 22	syl 14	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. -. ( ( G ` <. 1o , (/) >. ) = (/) \/ ( G ` <. 1o , (/) >. ) = { (/) } ) )
24		ioran 752	. . . . . 6 |- ( -. ( ( G ` <. 1o , (/) >. ) = (/) \/ ( G ` <. 1o , (/) >. ) = { (/) } ) <-> ( -. ( G ` <. 1o , (/) >. ) = (/) /\ -. ( G ` <. 1o , (/) >. ) = { (/) } ) )
25	23, 24	sylnib 676	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. ( -. ( G ` <. 1o , (/) >. ) = (/) /\ -. ( G ` <. 1o , (/) >. ) = { (/) } ) )
26		1n0 6435	. . . . . . . . . . 11 |- 1o =/= (/)
27	26	neii 2349	. . . . . . . . . 10 |- -. 1o = (/)
28		1oex 6427	. . . . . . . . . . 11 |- 1o e. _V
29		0ex 4132	. . . . . . . . . . 11 |- (/) e. _V
30	28, 29	opth1 4238	. . . . . . . . . 10 |- ( <. 1o , (/) >. = <. (/) , (/) >. -> 1o = (/) )
31	27, 30	mto 662	. . . . . . . . 9 |- -. <. 1o , (/) >. = <. (/) , (/) >.
32		0lt2o 6444	. . . . . . . . . . . . . 14 |- (/) e. 2o
33	6, 32	jctil 312	. . . . . . . . . . . . 13 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( (/) e. 2o /\ 2o e. N ) )
34		ontr1 4391	. . . . . . . . . . . . 13 |- ( N e. On -> ( ( (/) e. 2o /\ 2o e. N ) -> (/) e. N ) )
35	5, 33, 34	sylc 62	. . . . . . . . . . . 12 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> (/) e. N )
36		opelxpi 4660	. . . . . . . . . . . 12 |- ( ( (/) e. N /\ (/) e. 1o ) -> <. (/) , (/) >. e. ( N X. 1o ) )
37	35, 11, 36	sylancl 413	. . . . . . . . . . 11 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> <. (/) , (/) >. e. ( N X. 1o ) )
38	37, 16	eleqtrrdi 2271	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> <. (/) , (/) >. e. T )
39		f1veqaeq 5772	. . . . . . . . . 10 |- ( ( G : T -1-1-> ~P 1o /\ ( <. 1o , (/) >. e. T /\ <. (/) , (/) >. e. T ) ) -> ( ( G ` <. 1o , (/) >. ) = ( G ` <. (/) , (/) >. ) -> <. 1o , (/) >. = <. (/) , (/) >. ) )
40	1, 17, 38, 39	syl12anc 1236	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( ( G ` <. 1o , (/) >. ) = ( G ` <. (/) , (/) >. ) -> <. 1o , (/) >. = <. (/) , (/) >. ) )
41	31, 40	mtoi 664	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. ( G ` <. 1o , (/) >. ) = ( G ` <. (/) , (/) >. ) )
42	41	adantr 276	. . . . . . 7 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) -> -. ( G ` <. 1o , (/) >. ) = ( G ` <. (/) , (/) >. ) )
43		simpr 110	. . . . . . . 8 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) -> ( G ` <. (/) , (/) >. ) = (/) )
44	43	eqeq2d 2189	. . . . . . 7 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) -> ( ( G ` <. 1o , (/) >. ) = ( G ` <. (/) , (/) >. ) <-> ( G ` <. 1o , (/) >. ) = (/) ) )
45	42, 44	mtbid 672	. . . . . 6 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) -> -. ( G ` <. 1o , (/) >. ) = (/) )
46		2on0 6429	. . . . . . . . . . . . . 14 |- 2o =/= (/)
47	46	nesymi 2393	. . . . . . . . . . . . 13 |- -. (/) = 2o
48	29, 29	opth1 4238	. . . . . . . . . . . . 13 |- ( <. (/) , (/) >. = <. 2o , (/) >. -> (/) = 2o )
49	47, 48	mto 662	. . . . . . . . . . . 12 |- -. <. (/) , (/) >. = <. 2o , (/) >.
50		opelxpi 4660	. . . . . . . . . . . . . . 15 |- ( ( 2o e. N /\ (/) e. 1o ) -> <. 2o , (/) >. e. ( N X. 1o ) )
51	6, 11, 50	sylancl 413	. . . . . . . . . . . . . 14 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> <. 2o , (/) >. e. ( N X. 1o ) )
52	51, 16	eleqtrrdi 2271	. . . . . . . . . . . . 13 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> <. 2o , (/) >. e. T )
53		f1veqaeq 5772	. . . . . . . . . . . . 13 |- ( ( G : T -1-1-> ~P 1o /\ ( <. (/) , (/) >. e. T /\ <. 2o , (/) >. e. T ) ) -> ( ( G ` <. (/) , (/) >. ) = ( G ` <. 2o , (/) >. ) -> <. (/) , (/) >. = <. 2o , (/) >. ) )
54	1, 38, 52, 53	syl12anc 1236	. . . . . . . . . . . 12 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( ( G ` <. (/) , (/) >. ) = ( G ` <. 2o , (/) >. ) -> <. (/) , (/) >. = <. 2o , (/) >. ) )
55	49, 54	mtoi 664	. . . . . . . . . . 11 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. ( G ` <. (/) , (/) >. ) = ( G ` <. 2o , (/) >. ) )
56	55	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> -. ( G ` <. (/) , (/) >. ) = ( G ` <. 2o , (/) >. ) )
57		simplr 528	. . . . . . . . . . 11 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> ( G ` <. (/) , (/) >. ) = (/) )
58	57	eqeq1d 2186	. . . . . . . . . 10 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> ( ( G ` <. (/) , (/) >. ) = ( G ` <. 2o , (/) >. ) <-> (/) = ( G ` <. 2o , (/) >. ) ) )
59	56, 58	mtbid 672	. . . . . . . . 9 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> -. (/) = ( G ` <. 2o , (/) >. ) )
60		eqcom 2179	. . . . . . . . 9 |- ( (/) = ( G ` <. 2o , (/) >. ) <-> ( G ` <. 2o , (/) >. ) = (/) )
61	59, 60	sylnib 676	. . . . . . . 8 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> -. ( G ` <. 2o , (/) >. ) = (/) )
62		1onn 6523	. . . . . . . . . . . . . . . . . 18 |- 1o e. om
63		peano1 4595	. . . . . . . . . . . . . . . . . 18 |- (/) e. om
64		peano4 4598	. . . . . . . . . . . . . . . . . 18 |- ( ( 1o e. om /\ (/) e. om ) -> ( suc 1o = suc (/) <-> 1o = (/) ) )
65	62, 63, 64	mp2an 426	. . . . . . . . . . . . . . . . 17 |- ( suc 1o = suc (/) <-> 1o = (/) )
66	26, 65	nemtbir 2436	. . . . . . . . . . . . . . . 16 |- -. suc 1o = suc (/)
67		df-2o 6420	. . . . . . . . . . . . . . . . 17 |- 2o = suc 1o
68		df-1o 6419	. . . . . . . . . . . . . . . . 17 |- 1o = suc (/)
69	67, 68	eqeq12i 2191	. . . . . . . . . . . . . . . 16 |- ( 2o = 1o <-> suc 1o = suc (/) )
70	66, 69	mtbir 671	. . . . . . . . . . . . . . 15 |- -. 2o = 1o
71	70	neir 2350	. . . . . . . . . . . . . 14 |- 2o =/= 1o
72	71	nesymi 2393	. . . . . . . . . . . . 13 |- -. 1o = 2o
73	28, 29	opth1 4238	. . . . . . . . . . . . 13 |- ( <. 1o , (/) >. = <. 2o , (/) >. -> 1o = 2o )
74	72, 73	mto 662	. . . . . . . . . . . 12 |- -. <. 1o , (/) >. = <. 2o , (/) >.
75		f1veqaeq 5772	. . . . . . . . . . . . 13 |- ( ( G : T -1-1-> ~P 1o /\ ( <. 1o , (/) >. e. T /\ <. 2o , (/) >. e. T ) ) -> ( ( G ` <. 1o , (/) >. ) = ( G ` <. 2o , (/) >. ) -> <. 1o , (/) >. = <. 2o , (/) >. ) )
76	1, 17, 52, 75	syl12anc 1236	. . . . . . . . . . . 12 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( ( G ` <. 1o , (/) >. ) = ( G ` <. 2o , (/) >. ) -> <. 1o , (/) >. = <. 2o , (/) >. ) )
77	74, 76	mtoi 664	. . . . . . . . . . 11 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. ( G ` <. 1o , (/) >. ) = ( G ` <. 2o , (/) >. ) )
78	77	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> -. ( G ` <. 1o , (/) >. ) = ( G ` <. 2o , (/) >. ) )
79		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> ( G ` <. 1o , (/) >. ) = { (/) } )
80	79	eqeq1d 2186	. . . . . . . . . 10 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> ( ( G ` <. 1o , (/) >. ) = ( G ` <. 2o , (/) >. ) <-> { (/) } = ( G ` <. 2o , (/) >. ) ) )
81	78, 80	mtbid 672	. . . . . . . . 9 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> -. { (/) } = ( G ` <. 2o , (/) >. ) )
82		eqcom 2179	. . . . . . . . 9 |- ( { (/) } = ( G ` <. 2o , (/) >. ) <-> ( G ` <. 2o , (/) >. ) = { (/) } )
83	81, 82	sylnib 676	. . . . . . . 8 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> -. ( G ` <. 2o , (/) >. ) = { (/) } )
84	61, 83	jca 306	. . . . . . 7 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> ( -. ( G ` <. 2o , (/) >. ) = (/) /\ -. ( G ` <. 2o , (/) >. ) = { (/) } ) )
85	3, 52	ffvelcdmd 5654	. . . . . . . . . . . 12 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( G ` <. 2o , (/) >. ) e. ~P 1o )
86	85	elpwid 3588	. . . . . . . . . . 11 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( G ` <. 2o , (/) >. ) C_ 1o )
87	86, 20	sseqtrdi 3205	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( G ` <. 2o , (/) >. ) C_ { (/) } )
88		pwtrufal 14832	. . . . . . . . . 10 |- ( ( G ` <. 2o , (/) >. ) C_ { (/) } -> -. -. ( ( G ` <. 2o , (/) >. ) = (/) \/ ( G ` <. 2o , (/) >. ) = { (/) } ) )
89	87, 88	syl 14	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. -. ( ( G ` <. 2o , (/) >. ) = (/) \/ ( G ` <. 2o , (/) >. ) = { (/) } ) )
90		ioran 752	. . . . . . . . 9 |- ( -. ( ( G ` <. 2o , (/) >. ) = (/) \/ ( G ` <. 2o , (/) >. ) = { (/) } ) <-> ( -. ( G ` <. 2o , (/) >. ) = (/) /\ -. ( G ` <. 2o , (/) >. ) = { (/) } ) )
91	89, 90	sylnib 676	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. ( -. ( G ` <. 2o , (/) >. ) = (/) /\ -. ( G ` <. 2o , (/) >. ) = { (/) } ) )
92	91	ad2antrr 488	. . . . . . 7 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) /\ ( G ` <. 1o , (/) >. ) = { (/) } ) -> -. ( -. ( G ` <. 2o , (/) >. ) = (/) /\ -. ( G ` <. 2o , (/) >. ) = { (/) } ) )
93	84, 92	pm2.65da 661	. . . . . 6 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) -> -. ( G ` <. 1o , (/) >. ) = { (/) } )
94	45, 93	jca 306	. . . . 5 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = (/) ) -> ( -. ( G ` <. 1o , (/) >. ) = (/) /\ -. ( G ` <. 1o , (/) >. ) = { (/) } ) )
95	25, 94	mtand 665	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. ( G ` <. (/) , (/) >. ) = (/) )
96		eqcom 2179	. . . . . . . . . . 11 |- ( ( G ` <. 1o , (/) >. ) = ( G ` <. 2o , (/) >. ) <-> ( G ` <. 2o , (/) >. ) = ( G ` <. 1o , (/) >. ) )
97	77, 96	sylnib 676	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. ( G ` <. 2o , (/) >. ) = ( G ` <. 1o , (/) >. ) )
98	97	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> -. ( G ` <. 2o , (/) >. ) = ( G ` <. 1o , (/) >. ) )
99		simpr 110	. . . . . . . . . 10 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> ( G ` <. 1o , (/) >. ) = (/) )
100	99	eqeq2d 2189	. . . . . . . . 9 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> ( ( G ` <. 2o , (/) >. ) = ( G ` <. 1o , (/) >. ) <-> ( G ` <. 2o , (/) >. ) = (/) ) )
101	98, 100	mtbid 672	. . . . . . . 8 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> -. ( G ` <. 2o , (/) >. ) = (/) )
102	55	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> -. ( G ` <. (/) , (/) >. ) = ( G ` <. 2o , (/) >. ) )
103		eqcom 2179	. . . . . . . . . 10 |- ( ( G ` <. (/) , (/) >. ) = ( G ` <. 2o , (/) >. ) <-> ( G ` <. 2o , (/) >. ) = ( G ` <. (/) , (/) >. ) )
104	102, 103	sylnib 676	. . . . . . . . 9 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> -. ( G ` <. 2o , (/) >. ) = ( G ` <. (/) , (/) >. ) )
105		simplr 528	. . . . . . . . . 10 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> ( G ` <. (/) , (/) >. ) = { (/) } )
106	105	eqeq2d 2189	. . . . . . . . 9 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> ( ( G ` <. 2o , (/) >. ) = ( G ` <. (/) , (/) >. ) <-> ( G ` <. 2o , (/) >. ) = { (/) } ) )
107	104, 106	mtbid 672	. . . . . . . 8 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> -. ( G ` <. 2o , (/) >. ) = { (/) } )
108	101, 107	jca 306	. . . . . . 7 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> ( -. ( G ` <. 2o , (/) >. ) = (/) /\ -. ( G ` <. 2o , (/) >. ) = { (/) } ) )
109	91	ad2antrr 488	. . . . . . 7 |- ( ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) /\ ( G ` <. 1o , (/) >. ) = (/) ) -> -. ( -. ( G ` <. 2o , (/) >. ) = (/) /\ -. ( G ` <. 2o , (/) >. ) = { (/) } ) )
110	108, 109	pm2.65da 661	. . . . . 6 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) -> -. ( G ` <. 1o , (/) >. ) = (/) )
111	41	adantr 276	. . . . . . 7 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) -> -. ( G ` <. 1o , (/) >. ) = ( G ` <. (/) , (/) >. ) )
112		simpr 110	. . . . . . . 8 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) -> ( G ` <. (/) , (/) >. ) = { (/) } )
113	112	eqeq2d 2189	. . . . . . 7 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) -> ( ( G ` <. 1o , (/) >. ) = ( G ` <. (/) , (/) >. ) <-> ( G ` <. 1o , (/) >. ) = { (/) } ) )
114	111, 113	mtbid 672	. . . . . 6 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) -> -. ( G ` <. 1o , (/) >. ) = { (/) } )
115	110, 114	jca 306	. . . . 5 |- ( ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) /\ ( G ` <. (/) , (/) >. ) = { (/) } ) -> ( -. ( G ` <. 1o , (/) >. ) = (/) /\ -. ( G ` <. 1o , (/) >. ) = { (/) } ) )
116	25, 115	mtand 665	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. ( G ` <. (/) , (/) >. ) = { (/) } )
117	95, 116	jca 306	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( -. ( G ` <. (/) , (/) >. ) = (/) /\ -. ( G ` <. (/) , (/) >. ) = { (/) } ) )
118	3, 38	ffvelcdmd 5654	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( G ` <. (/) , (/) >. ) e. ~P 1o )
119	118	elpwid 3588	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( G ` <. (/) , (/) >. ) C_ 1o )
120	119, 20	sseqtrdi 3205	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> ( G ` <. (/) , (/) >. ) C_ { (/) } )
121		pwtrufal 14832	. . . . 5 |- ( ( G ` <. (/) , (/) >. ) C_ { (/) } -> -. -. ( ( G ` <. (/) , (/) >. ) = (/) \/ ( G ` <. (/) , (/) >. ) = { (/) } ) )
122	120, 121	syl 14	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. -. ( ( G ` <. (/) , (/) >. ) = (/) \/ ( G ` <. (/) , (/) >. ) = { (/) } ) )
123		ioran 752	. . . 4 |- ( -. ( ( G ` <. (/) , (/) >. ) = (/) \/ ( G ` <. (/) , (/) >. ) = { (/) } ) <-> ( -. ( G ` <. (/) , (/) >. ) = (/) /\ -. ( G ` <. (/) , (/) >. ) = { (/) } ) )
124	122, 123	sylnib 676	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ 2o e. N ) -> -. ( -. ( G ` <. (/) , (/) >. ) = (/) /\ -. ( G ` <. (/) , (/) >. ) = { (/) } ) )
125	117, 124	pm2.65da 661	. 2 |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> -. 2o e. N )
126		2onn 6524	. . . 4 |- 2o e. om
127		nntri1 6499	. . . 4 |- ( ( N e. om /\ 2o e. om ) -> ( N C_ 2o <-> -. 2o e. N ) )
128	126, 127	mpan2 425	. . 3 |- ( N e. om -> ( N C_ 2o <-> -. 2o e. N ) )
129	128	adantr 276	. 2 |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> ( N C_ 2o <-> -. 2o e. N ) )
130	125, 129	mpbird 167	1 |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> N C_ 2o )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   C_ wss 3131   (/)c0 3424   ~Pcpw 3577   {csn 3594   <.cop 3597   U_ciun 3888   Oncon0 4365   suc csuc 4367   omcom 4591   X. cxp 4626   -->wf 5214   -1-1->wf1 5215   ` cfv 5218   1oc1o 6412   2oc2o 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fv 5226  df-1o 6419  df-2o 6420

This theorem is referenced by:  pwf1oexmid  14834