Theorem 2omotaplemap 7324

Description: Lemma for 2omotap 7326. (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 4737	. . 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 4034	. . . . . . . 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 2380	. . . . . . . . . 10 |- ( u = a -> ( u =/= v <-> a =/= v ) )
5	4	anbi2d 464	. . . . . . . . 9 |- ( u = a -> ( ( ph /\ u =/= v ) <-> ( ph /\ a =/= v ) ) )
6		neeq2 2381	. . . . . . . . . 10 |- ( v = b -> ( a =/= v <-> a =/= b ) )
7	6	anbi2d 464	. . . . . . . . 9 |- ( v = b -> ( ( ph /\ a =/= v ) <-> ( ph /\ a =/= b ) ) )
8	5, 7	opelopab2 4305	. . . . . . . 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 4034	. . . . . . . 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 2380	. . . . . . . . . . . 12 |- ( u = b -> ( u =/= v <-> b =/= v ) )
12	11	anbi2d 464	. . . . . . . . . . 11 |- ( u = b -> ( ( ph /\ u =/= v ) <-> ( ph /\ b =/= v ) ) )
13		neeq2 2381	. . . . . . . . . . . 12 |- ( v = a -> ( b =/= v <-> b =/= a ) )
14	13	anbi2d 464	. . . . . . . . . . 11 |- ( v = a -> ( ( ph /\ b =/= v ) <-> ( ph /\ b =/= a ) ) )
15	12, 14	opelopab2 4305	. . . . . . . . . 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 2451	. . . . . . . . . 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 2583	. . . 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 2376	. . . . . 6 |- -. a =/= a
26	25	intnan 930	. . . . 5 |- -. ( ph /\ a =/= a )
27		df-br 4034	. . . . . 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 2381	. . . . . . . . 9 |- ( v = a -> ( a =/= v <-> a =/= a ) )
29	28	anbi2d 464	. . . . . . . 8 |- ( v = a -> ( ( ph /\ a =/= v ) <-> ( ph /\ a =/= a ) ) )
30	5, 29	opelopab2 4305	. . . . . . 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 676	. . . 4 |- ( a e. 2o -> -. a { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } a )
34	33	rgen 2550	. . 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 1019	. . . . . 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 2393	. . . . . . . . . . 11 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ a = c ) -> c =/= b )
40	39	necomd 2453	. . . . . . . . . 10 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ a = c ) -> b =/= c )
41	40	olcd 735	. . . . . . . . 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 2374	. . . . . . . . . 10 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ -. a = c ) -> a =/= c )
44	43	orcd 734	. . . . . . . . 9 |- ( ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) /\ -. a = c ) -> ( a =/= c \/ b =/= c ) )
45		simpl1 1002	. . . . . . . . . . . 12 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> a e. 2o )
46		2onn 6579	. . . . . . . . . . . 12 |- 2o e. om
47		elnn 4642	. . . . . . . . . . . 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 1004	. . . . . . . . . . . 12 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> c e. 2o )
50		elnn 4642	. . . . . . . . . . . 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 6557	. . . . . . . . . . 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 837	. . . . . . . . . 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 799	. . . . . . . 8 |- ( ( ( a e. 2o /\ b e. 2o /\ c e. 2o ) /\ ( ph /\ a =/= b ) ) -> ( a =/= c \/ b =/= c ) )
57		df-br 4034	. . . . . . . . . . . 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 2381	. . . . . . . . . . . . . . 15 |- ( v = c -> ( a =/= v <-> a =/= c ) )
59	58	anbi2d 464	. . . . . . . . . . . . . 14 |- ( v = c -> ( ( ph /\ a =/= v ) <-> ( ph /\ a =/= c ) ) )
60	5, 59	opelopab2 4305	. . . . . . . . . . . . 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 1018	. . . . . . . . . . . 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 4034	. . . . . . . . . . . 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 2381	. . . . . . . . . . . . . . 15 |- ( v = c -> ( b =/= v <-> b =/= c ) )
70	69	anbi2d 464	. . . . . . . . . . . . . 14 |- ( v = c -> ( ( ph /\ b =/= v ) <-> ( ph /\ b =/= c ) ) )
71	12, 70	opelopab2 4305	. . . . . . . . . . . . 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 1017	. . . . . . . . . . . 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 794	. . . . . . . 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 2580	. . 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 668	. . . . . 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 677	. . . . . . . . 9 |- ( ( ( -. -. ph /\ ( a e. 2o /\ b e. 2o ) ) /\ -. ( ph /\ a =/= b ) ) -> -. ( a =/= b /\ ph ) )
91		imnan 691	. . . . . . . . 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 664	. . . . . . 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 4642	. . . . . . . . . 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 6557	. . . . . . . . 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 2372	. . . . . . . 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 2579	. . 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 7318	. 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 1183	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 709  DECID wdc 835   /\ w3a 980   e. wcel 2167   =/= wne 2367   A.wral 2475   C_ wss 3157   <.cop 3625   class class class wbr 4033   {copab 4093   omcom 4626   X. cxp 4661   2oc2o 6468   TAp wtap 7316

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-1o 6474  df-2o 6475  df-pap 7315  df-tap 7317

This theorem is referenced by:  2omotaplemst  7325