Theorem 2omotaplemap 7256

Description: Lemma for 2omotap 7258. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2omotaplemap	|- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )

Distinct variable group:   ph, u, v

Proof of Theorem 2omotaplemap

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabssxp 4701	. . 3 |- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } C_ ( 2o X. 2o )
2	1	a1i 9	. 2 |- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } C_ ( 2o X. 2o ) )
3		df-br 4005	. . . . . . . 8 |- ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b <-> <. a , b >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
4		neeq1 2360	. . . . . . . . . 10 |- ( u = a -> ( u =/= v <-> a =/= v ) )
5	4	anbi2d 464	. . . . . . . . 9 |- ( u = a -> ( ( ph /\ u =/= v ) <-> ( ph /\ a =/= v ) ) )
6		neeq2 2361	. . . . . . . . . 10 |- ( v = b -> ( a =/= v <-> a =/= b ) )
7	6	anbi2d 464	. . . . . . . . 9 |- ( v = b -> ( ( ph /\ a =/= v ) <-> ( ph /\ a =/= b ) ) )
8	5, 7	opelopab2 4271	. . . . . . . 8 |- ( ( a e. 2o /\ b e. 2o ) -> ( <. a , b >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ a =/= b ) ) )
9	3, 8	bitrid 192	. . . . . . 7 |- ( ( a e. 2o /\ b e. 2o ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b <-> ( ph /\ a =/= b ) ) )
10		df-br 4005	. . . . . . . 8 |- ( b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a <-> <. b , a >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
11		neeq1 2360	. . . . . . . . . . . 12 |- ( u = b -> ( u =/= v <-> b =/= v ) )
12	11	anbi2d 464	. . . . . . . . . . 11 |- ( u = b -> ( ( ph /\ u =/= v ) <-> ( ph /\ b =/= v ) ) )
13		neeq2 2361	. . . . . . . . . . . 12 |- ( v = a -> ( b =/= v <-> b =/= a ) )
14	13	anbi2d 464	. . . . . . . . . . 11 |- ( v = a -> ( ( ph /\ b =/= v ) <-> ( ph /\ b =/= a ) ) )
15	12, 14	opelopab2 4271	. . . . . . . . . 10 |- ( ( b e. 2o /\ a e. 2o ) -> ( <. b , a >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ b =/= a ) ) )
16	15	ancoms 268	. . . . . . . . 9 |- ( ( a e. 2o /\ b e. 2o ) -> ( <. b , a >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ b =/= a ) ) )
17		necom 2431	. . . . . . . . . 10 |- ( b =/= a <-> a =/= b )
18	17	anbi2i 457	. . . . . . . . 9 |- ( ( ph /\ b =/= a ) <-> ( ph /\ a =/= b ) )
19	16, 18	bitrdi 196	. . . . . . . 8 |- ( ( a e. 2o /\ b e. 2o ) -> ( <. b , a >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ a =/= b ) ) )
20	10, 19	bitrid 192	. . . . . . 7 |- ( ( a e. 2o /\ b e. 2o ) -> ( b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a <-> ( ph /\ a =/= b ) ) )
21	9, 20	bitr4d 191	. . . . . 6 |- ( ( a e. 2o /\ b e. 2o ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b <-> b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a ) )
22	21	biimpd 144	. . . . 5 |- ( ( a e. 2o /\ b e. 2o ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a ) )
23	22	rgen2 2563	. . . 4 |- A. a e. 2o A. b e. 2o ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a )
24	23	a1i 9	. . 3 |- ( -. -. ph -> A. a e. 2o A. b e. 2o ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a ) )
25		neirr 2356	. . . . . 6 |- -. a =/= a
26	25	intnan 929	. . . . 5 |- -. ( ph /\ a =/= a )
27		df-br 4005	. . . . . 6 |- ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a <-> <. a , a >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
28		neeq2 2361	. . . . . . . . 9 |- ( v = a -> ( a =/= v <-> a =/= a ) )
29	28	anbi2d 464	. . . . . . . 8 |- ( v = a -> ( ( ph /\ a =/= v ) <-> ( ph /\ a =/= a ) ) )
30	5, 29	opelopab2 4271	. . . . . . 7 |- ( ( a e. 2o /\ a e. 2o ) -> ( <. a , a >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ a =/= a ) ) )
31	30	anidms 397	. . . . . 6 |- ( a e. 2o -> ( <. a , a >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ a =/= a ) ) )
32	27, 31	bitrid 192	. . . . 5 |- ( a e. 2o -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a <-> ( ph /\ a =/= a ) ) )
33	26, 32	mtbiri 675	. . . 4 |- ( a e. 2o -> -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a )
34	33	rgen 2530	. . 3 |- A. a e. 2o -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a
35	24, 34	jctil 312	. 2 |- ( -. -. ph -> ( A. a e. 2o -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a /\ A. a e. 2o A. b e. 2o ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a ) ) )
36	9	3adant3 1017	. . . . . 6 |- ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b <-> ( ph /\ a =/= b ) ) )
37		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ a = c ) -> a = c )
38		simplrr 536	. . . . . . . . . . . 12 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ a = c ) -> a =/= b )
39	37, 38	eqnetrrd 2373	. . . . . . . . . . 11 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ a = c ) -> c =/= b )
40	39	necomd 2433	. . . . . . . . . 10 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ a = c ) -> b =/= c )
41	40	olcd 734	. . . . . . . . 9 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ a = c ) -> ( a =/= c \/ b =/= c ) )
42		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ -. a = c ) -> -. a = c )
43	42	neqned 2354	. . . . . . . . . 10 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ -. a = c ) -> a =/= c )
44	43	orcd 733	. . . . . . . . 9 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ -. a = c ) -> ( a =/= c \/ b =/= c ) )
45		simpl1 1000	. . . . . . . . . . . 12 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> a e. 2o )
46		2onn 6522	. . . . . . . . . . . 12 |- 2o e. om
47		elnn 4606	. . . . . . . . . . . 12 |- ( ( a e. 2o /\ 2o e. om ) -> a e. om )
48	45, 46, 47	sylancl 413	. . . . . . . . . . 11 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> a e. om )
49		simpl3 1002	. . . . . . . . . . . 12 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> c e. 2o )
50		elnn 4606	. . . . . . . . . . . 12 |- ( ( c e. 2o /\ 2o e. om ) -> c e. om )
51	49, 46, 50	sylancl 413	. . . . . . . . . . 11 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> c e. om )
52		nndceq 6500	. . . . . . . . . . 11 |- ( ( a e. om /\ c e. om ) -> DECID a = c )
53	48, 51, 52	syl2anc 411	. . . . . . . . . 10 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> DECID a = c )
54		exmiddc 836	. . . . . . . . . 10 |- (DECID a = c -> ( a = c \/ -. a = c ) )
55	53, 54	syl 14	. . . . . . . . 9 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( a = c \/ -. a = c ) )
56	41, 44, 55	mpjaodan 798	. . . . . . . 8 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( a =/= c \/ b =/= c ) )
57		df-br 4005	. . . . . . . . . . . 12 |- ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c <-> <. a , c >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
58		neeq2 2361	. . . . . . . . . . . . . . 15 |- ( v = c -> ( a =/= v <-> a =/= c ) )
59	58	anbi2d 464	. . . . . . . . . . . . . 14 |- ( v = c -> ( ( ph /\ a =/= v ) <-> ( ph /\ a =/= c ) ) )
60	5, 59	opelopab2 4271	. . . . . . . . . . . . 13 |- ( ( a e. 2o /\ c e. 2o ) -> ( <. a , c >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ a =/= c ) ) )
61	60	3adant2 1016	. . . . . . . . . . . 12 |- ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) -> ( <. a , c >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ a =/= c ) ) )
62	57, 61	bitrid 192	. . . . . . . . . . 11 |- ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c <-> ( ph /\ a =/= c ) ) )
63	62	adantr 276	. . . . . . . . . 10 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c <-> ( ph /\ a =/= c ) ) )
64		ibar 301	. . . . . . . . . . . 12 |- ( ph -> ( a =/= c <-> ( ph /\ a =/= c ) ) )
65	64	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ a =/= b ) -> ( a =/= c <-> ( ph /\ a =/= c ) ) )
66	65	adantl 277	. . . . . . . . . 10 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( a =/= c <-> ( ph /\ a =/= c ) ) )
67	63, 66	bitr4d 191	. . . . . . . . 9 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c <-> a =/= c ) )
68		df-br 4005	. . . . . . . . . . . 12 |- ( b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c <-> <. b , c >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } )
69		neeq2 2361	. . . . . . . . . . . . . . 15 |- ( v = c -> ( b =/= v <-> b =/= c ) )
70	69	anbi2d 464	. . . . . . . . . . . . . 14 |- ( v = c -> ( ( ph /\ b =/= v ) <-> ( ph /\ b =/= c ) ) )
71	12, 70	opelopab2 4271	. . . . . . . . . . . . 13 |- ( ( b e. 2o /\ c e. 2o ) -> ( <. b , c >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ b =/= c ) ) )
72	71	3adant1 1015	. . . . . . . . . . . 12 |- ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) -> ( <. b , c >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } <-> ( ph /\ b =/= c ) ) )
73	68, 72	bitrid 192	. . . . . . . . . . 11 |- ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) -> ( b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c <-> ( ph /\ b =/= c ) ) )
74	73	adantr 276	. . . . . . . . . 10 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c <-> ( ph /\ b =/= c ) ) )
75		ibar 301	. . . . . . . . . . . 12 |- ( ph -> ( b =/= c <-> ( ph /\ b =/= c ) ) )
76	75	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ a =/= b ) -> ( b =/= c <-> ( ph /\ b =/= c ) ) )
77	76	adantl 277	. . . . . . . . . 10 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( b =/= c <-> ( ph /\ b =/= c ) ) )
78	74, 77	bitr4d 191	. . . . . . . . 9 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c <-> b =/= c ) )
79	67, 78	orbi12d 793	. . . . . . . 8 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c \/ b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c ) <-> ( a =/= c \/ b =/= c ) ) )
80	56, 79	mpbird 167	. . . . . . 7 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c \/ b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c ) )
81	80	ex 115	. . . . . 6 |- ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) -> ( ( ph /\ a =/= b ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c \/ b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c ) ) )
82	36, 81	sylbid 150	. . . . 5 |- ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c \/ b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c ) ) )
83	82	adantl 277	. . . 4 |- ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o /\ c e. 2o ) ) -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c \/ b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c ) ) )
84	83	ralrimivvva 2560	. . 3 |- ( -. -. ph -> A. a e. 2o A. b e. 2o A. c e. 2o ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c \/ b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c ) ) )
85	9	notbid 667	. . . . . 6 |- ( ( a e. 2o /\ b e. 2o ) -> ( -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b <-> -. ( ph /\ a =/= b ) ) )
86	85	adantl 277	. . . . 5 |- ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) -> ( -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b <-> -. ( ph /\ a =/= b ) ) )
87		simpll 527	. . . . . . . 8 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> -. -. ph )
88		simpr 110	. . . . . . . . . 10 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> -. ( ph /\ a =/= b ) )
89		ancom 266	. . . . . . . . . 10 |- ( ( ph /\ a =/= b ) <-> ( a =/= b /\ ph ) )
90	88, 89	sylnib 676	. . . . . . . . 9 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> -. ( a =/= b /\ ph ) )
91		imnan 690	. . . . . . . . 9 |- ( ( a =/= b -> -. ph ) <-> -. ( a =/= b /\ ph ) )
92	90, 91	sylibr 134	. . . . . . . 8 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> ( a =/= b -> -. ph ) )
93	87, 92	mtod 663	. . . . . . 7 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> -. a =/= b )
94		simplrl 535	. . . . . . . . . 10 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> a e. 2o )
95	94, 46, 47	sylancl 413	. . . . . . . . 9 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> a e. om )
96		simplrr 536	. . . . . . . . . 10 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> b e. 2o )
97		elnn 4606	. . . . . . . . . 10 |- ( ( b e. 2o /\ 2o e. om ) -> b e. om )
98	96, 46, 97	sylancl 413	. . . . . . . . 9 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> b e. om )
99		nndceq 6500	. . . . . . . . 9 |- ( ( a e. om /\ b e. om ) -> DECID a = b )
100	95, 98, 99	syl2anc 411	. . . . . . . 8 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> DECID a = b )
101		nnedc 2352	. . . . . . . 8 |- (DECID a = b -> ( -. a =/= b <-> a = b ) )
102	100, 101	syl 14	. . . . . . 7 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> ( -. a =/= b <-> a = b ) )
103	93, 102	mpbid 147	. . . . . 6 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> a = b )
104	103	ex 115	. . . . 5 |- ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) -> ( -. ( ph /\ a =/= b ) -> a = b ) )
105	86, 104	sylbid 150	. . . 4 |- ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) -> ( -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> a = b ) )
106	105	ralrimivva 2559	. . 3 |- ( -. -. ph -> A. a e. 2o A. b e. 2o ( -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> a = b ) )
107	84, 106	jca 306	. 2 |- ( -. -. ph -> ( A. a e. 2o A. b e. 2o A. c e. 2o ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c \/ b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c ) ) /\ A. a e. 2o A. b e. 2o ( -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> a = b ) ) )
108		dftap2 7250	. 2 |- ( { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o <-> ( { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } C_ ( 2o X. 2o ) /\ ( A. a e. 2o -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a /\ A. a e. 2o A. b e. 2o ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a ) ) /\ ( A. a e. 2o A. b e. 2o A. c e. 2o ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> ( a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c \/ b { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } c ) ) /\ A. a e. 2o A. b e. 2o ( -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } b -> a = b ) ) ) )
109	2, 35, 107, 108	syl3anbrc 1181	1 |- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   /\ w3a 978   e. wcel 2148   =/= wne 2347   A.wral 2455   C_ wss 3130   <.cop 3596   class class class wbr 4004   {copab 4064   omcom 4590   X. cxp 4625   2oc2o 6411   TAp wtap 7248

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 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  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 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-1o 6417  df-2o 6418  df-pap 7247  df-tap 7249

This theorem is referenced by:  2omotaplemst  7257