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Theorem nchoicelem16 6304
Description: Lemma for nchoice 6308. Set up stratification for nchoicelem17 6305. (Contributed by SF, 19-Mar-2015.)
Assertion
Ref Expression
nchoicelem16 <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
Distinct variable group:   ,
Proof of Theorem nchoicelem16
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unab 3521 . . 3 <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
2 elima3 4756 . . . . . 6 1c AddC ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC 1c AddC
3 vex 2862 . . . . . . . . . . 11
43eluni1 4173 . . . . . . . . . 10 ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC
5 snex 4111 . . . . . . . . . . 11
65elcompl 3225 . . . . . . . . . 10 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC
7 elimapw13 4946 . . . . . . . . . . . 12 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC NC SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c
8 spacval 6282 . . . . . . . . . . . . . . . . . 18 NC Spac Clos1 NC NC 2c ↑c
98nceqd 6110 . . . . . . . . . . . . . . . . 17 NC Nc Spac Nc Clos1 NC NC 2c ↑c
109eqeq1d 2361 . . . . . . . . . . . . . . . 16 NC Nc Spac Nc Clos1 NC NC 2c ↑c
11 tccl 6160 . . . . . . . . . . . . . . . . . . . 20 NC Tc NC
12 spacval 6282 . . . . . . . . . . . . . . . . . . . 20 Tc NC Spac Tc Clos1 Tc NC NC 2c ↑c
1311, 12syl 15 . . . . . . . . . . . . . . . . . . 19 NC Spac Tc Clos1 Tc NC NC 2c ↑c
1413eleq1d 2419 . . . . . . . . . . . . . . . . . 18 NC Spac Tc Fin Clos1 Tc NC NC 2c ↑c Fin
1513nceqd 6110 . . . . . . . . . . . . . . . . . . . 20 NC Nc Spac Tc Nc Clos1 Tc NC NC 2c ↑c
1615eqeq1d 2361 . . . . . . . . . . . . . . . . . . 19 NC Nc Spac Tc Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 1c
1715eqeq1d 2361 . . . . . . . . . . . . . . . . . . 19 NC Nc Spac Tc Tc 2c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
1816, 17orbi12d 690 . . . . . . . . . . . . . . . . . 18 NC Nc Spac Tc Tc 1c Nc Spac Tc Tc 2c Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
1914, 18anbi12d 691 . . . . . . . . . . . . . . . . 17 NC Spac Tc Fin Nc Spac Tc Tc 1c Nc Spac Tc Tc 2c Clos1 Tc NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
2019notbid 285 . . . . . . . . . . . . . . . 16 NC Spac Tc Fin Nc Spac Tc Tc 1c Nc Spac Tc Tc 2c Clos1 Tc NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
2110, 20anbi12d 691 . . . . . . . . . . . . . . 15 NC Nc Spac Spac Tc Fin Nc Spac Tc Tc 1c Nc Spac Tc Tc 2c Nc Clos1 NC NC 2c ↑c Clos1 Tc NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
22 eldif 3221 . . . . . . . . . . . . . . . 16 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c
23 snex 4111 . . . . . . . . . . . . . . . . . . . 20
2423, 3opsnelsi 5774 . . . . . . . . . . . . . . . . . . 19 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
25 opelcnv 4893 . . . . . . . . . . . . . . . . . . 19 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
26 opelres 4950 . . . . . . . . . . . . . . . . . . . 20 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
27 opelcnv 4893 . . . . . . . . . . . . . . . . . . . . . 22 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
28 opelco 4884 . . . . . . . . . . . . . . . . . . . . . . 23 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
29 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3029brsnsi1 5775 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
3130anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
32 19.41v 1901 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
3331, 32bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . 25 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
3433exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . 24 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
35 excom 1741 . . . . . . . . . . . . . . . . . . . . . . . 24 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
3634, 35bitri 240 . . . . . . . . . . . . . . . . . . . . . . 23 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
37 anass 630 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
3837exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . 25 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
39 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . 26
40 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 S S
4140anbi2d 684 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
4239, 41ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . 25 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
43 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
44 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
45 spacvallem1 6281 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 NC NC 2c ↑c
4645, 29nchoicelem10 6298 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
4743, 44, 463bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
48 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4948, 3brssetsn 4759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 S
5047, 49anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . 25 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
5138, 42, 503bitri 262 . . . . . . . . . . . . . . . . . . . . . . . 24 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
5251exbii 1582 . . . . . . . . . . . . . . . . . . . . . . 23 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
5328, 36, 523bitri 262 . . . . . . . . . . . . . . . . . . . . . 22 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
5429, 45clos1ex 5876 . . . . . . . . . . . . . . . . . . . . . . 23 Clos1 NC NC 2c ↑c
55 eleq1 2413 . . . . . . . . . . . . . . . . . . . . . . 23 Clos1 NC NC 2c ↑c Clos1 NC NC 2c ↑c
5654, 55ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . 22 Clos1 NC NC 2c ↑c Clos1 NC NC 2c ↑c
5727, 53, 563bitri 262 . . . . . . . . . . . . . . . . . . . . 21 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
5857anbi1i 676 . . . . . . . . . . . . . . . . . . . 20 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC Clos1 NC NC 2c ↑c NC
59 ancom 437 . . . . . . . . . . . . . . . . . . . 20 Clos1 NC NC 2c ↑c NC NC Clos1 NC NC 2c ↑c
6026, 58, 593bitri 262 . . . . . . . . . . . . . . . . . . 19 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC NC Clos1 NC NC 2c ↑c
6124, 25, 603bitri 262 . . . . . . . . . . . . . . . . . 18 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC NC Clos1 NC NC 2c ↑c
62 eqcom 2355 . . . . . . . . . . . . . . . . . . 19 Nc Clos1 NC NC 2c ↑c Nc Clos1 NC NC 2c ↑c
6354eqnc2 6129 . . . . . . . . . . . . . . . . . . 19 Nc Clos1 NC NC 2c ↑c NC Clos1 NC NC 2c ↑c
6462, 63bitri 240 . . . . . . . . . . . . . . . . . 18 Nc Clos1 NC NC 2c ↑c NC Clos1 NC NC 2c ↑c
6561, 64bitr4i 243 . . . . . . . . . . . . . . . . 17 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC Nc Clos1 NC NC 2c ↑c
66 elimapw11c 4948 . . . . . . . . . . . . . . . . . . 19 SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
67 oteltxp 5782 . . . . . . . . . . . . . . . . . . . . 21 SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
68 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . 24
6968, 23opsnelsi 5774 . . . . . . . . . . . . . . . . . . . . . . 23 SI SI TcFn SI TcFn
70 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . 25
7170, 29opsnelsi 5774 . . . . . . . . . . . . . . . . . . . . . . . 24 SI TcFn TcFn
72 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . 24 TcFn TcFn
73 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . 24 TcFn TcFn
7471, 72, 733bitr2i 264 . . . . . . . . . . . . . . . . . . . . . . 23 SI TcFn TcFn
75 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . 24
7675brtcfn 6246 . . . . . . . . . . . . . . . . . . . . . . 23 TcFn Tc
7769, 74, 763bitri 262 . . . . . . . . . . . . . . . . . . . . . 22 SI SI TcFn Tc
78 opelco 4884 . . . . . . . . . . . . . . . . . . . . . . 23 TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC TcFn AddC 1c AddC 2c Nn
79 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . 26 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
80 brres 4949 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
81 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
82 brco 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
8368brsnsi1 5775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
8483anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
85 19.41v 1901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
8684, 85bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
8786exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
88 excom 1741 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
89 anass 630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
9089exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
91 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
92 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 S S
9392anbi2d 684 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
9491, 93ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
95 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
96 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
9745, 68nchoicelem10 6298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
9895, 96, 973bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
99 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
10099, 48brssetsn 4759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 S
10198, 100anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
10290, 94, 1013bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
103102exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
10487, 88, 1033bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
105 df-clel 2349 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Clos1 NC NC 2c ↑c Clos1 NC NC 2c ↑c
106104, 105bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
10781, 82, 1063bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
108107anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC Clos1 NC NC 2c ↑c NC
109 ancom 437 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Clos1 NC NC 2c ↑c NC NC Clos1 NC NC 2c ↑c
11080, 108, 1093bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC NC Clos1 NC NC 2c ↑c
11168, 45clos1ex 5876 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Clos1 NC NC 2c ↑c
112111eqnc2 6129 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Nc Clos1 NC NC 2c ↑c NC Clos1 NC NC 2c ↑c
113110, 112bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC Nc Clos1 NC NC 2c ↑c
11479, 113bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . 25 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC Nc Clos1 NC NC 2c ↑c
115 brres 4949 . . . . . . . . . . . . . . . . . . . . . . . . . 26 TcFn AddC 1c AddC 2c Nn TcFn AddC 1c AddC 2c Nn
116 brco 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 TcFn AddC 1c AddC 2c AddC 1c AddC 2c TcFn
117 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 AddC 1c AddC 2c AddC 1c AddC 2c
118 brun 4692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 AddC 1c AddC 2c AddC 1c AddC 2c
119 brco 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 AddC 1c 1c AddC
120 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1c 1c
121 brres 4949 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1c 1c
122 eliniseg 5020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1c 1c
123122anbi2i 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1c 1c
124121, 123bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1c 1c
125 1cex 4142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1c
12699, 125op1st2nd 5790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1c 1c
127120, 124, 1263bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1c 1c
128127anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1c AddC 1c AddC
129128exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1c AddC 1c AddC
13099, 125opex 4588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1c
131 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1c AddC 1c AddC
132130, 131ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1c AddC 1c AddC
133119, 129, 1323bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 AddC 1c 1c AddC
13499, 125braddcfn 5826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1c AddC 1c
135 eqcom 2355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1c 1c
136133, 134, 1353bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 AddC 1c 1c
137 brco 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 AddC 2c 2c AddC
138 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2c 2c
139 brres 4949 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2c 2c
140 eliniseg 5020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2c 2c
141140anbi2i 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2c 2c
142139, 141bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2c 2c
143 2nc 6168 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2c NC
144143elexi 2868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2c
14599, 144op1st2nd 5790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2c 2c
146138, 142, 1453bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2c 2c
147146anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2c AddC 2c AddC
148147exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2c AddC 2c AddC
14999, 144opex 4588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2c
150 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2c AddC 2c AddC
151149, 150ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2c AddC 2c AddC
152137, 148, 1513bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 AddC 2c 2c AddC
15399, 144braddcfn 5826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2c AddC 2c
154 eqcom 2355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2c 2c
155152, 153, 1543bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 AddC 2c 2c
156136, 155orbi12i 507 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 AddC 1c AddC 2c 1c 2c
157117, 118, 1563bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 AddC 1c AddC 2c 1c 2c
158 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 TcFn TcFn
1593brtcfn 6246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 TcFn Tc
160158, 159bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 TcFn Tc
161157, 160anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 AddC 1c AddC 2c TcFn 1c 2c Tc
162 ancom 437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1c 2c Tc Tc 1c 2c
163161, 162bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 AddC 1c AddC 2c TcFn Tc 1c 2c
164163exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 AddC 1c AddC 2c TcFn Tc 1c 2c
165 tcex 6157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Tc
166 addceq1 4383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Tc 1c Tc 1c
167166eqeq2d 2364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Tc 1c Tc 1c
168 addceq1 4383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Tc 2c Tc 2c
169168eqeq2d 2364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Tc 2c Tc 2c
170167, 169orbi12d 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Tc 1c 2c Tc 1c Tc 2c
171165, 170ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Tc 1c 2c Tc 1c Tc 2c
172116, 164, 1713bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 TcFn AddC 1c AddC 2c Tc 1c Tc 2c
173172anbi1i 676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 TcFn AddC 1c AddC 2c Nn Tc 1c Tc 2c Nn
174 ancom 437 . . . . . . . . . . . . . . . . . . . . . . . . . 26 Tc 1c Tc 2c Nn Nn Tc 1c Tc 2c
175115, 173, 1743bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . 25 TcFn AddC 1c AddC 2c Nn Nn Tc 1c Tc 2c
176114, 175anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . 24 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC TcFn AddC 1c AddC 2c Nn Nc Clos1 NC NC 2c ↑c Nn Tc 1c Tc 2c
177176exbii 1582 . . . . . . . . . . . . . . . . . . . . . . 23 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC TcFn AddC 1c AddC 2c Nn Nc Clos1 NC NC 2c ↑c Nn Tc 1c Tc 2c
178 ncex 6117 . . . . . . . . . . . . . . . . . . . . . . . 24 Nc Clos1 NC NC 2c ↑c
179 eleq1 2413 . . . . . . . . . . . . . . . . . . . . . . . . . 26 Nc Clos1 NC NC 2c ↑c Nn Nc Clos1 NC NC 2c ↑c Nn
180 finnc 6243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 Clos1 NC NC 2c ↑c Fin Nc Clos1 NC NC 2c ↑c Nn
181179, 180syl6bbr 254 . . . . . . . . . . . . . . . . . . . . . . . . 25 Nc Clos1 NC NC 2c ↑c Nn Clos1 NC NC 2c ↑c Fin
182 eqeq1 2359 . . . . . . . . . . . . . . . . . . . . . . . . . 26 Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 1c
183 eqeq1 2359 . . . . . . . . . . . . . . . . . . . . . . . . . 26 Nc Clos1 NC NC 2c ↑c Tc 2c Nc Clos1 NC NC 2c ↑c Tc 2c
184182, 183orbi12d 690 . . . . . . . . . . . . . . . . . . . . . . . . 25 Nc Clos1 NC NC 2c ↑c Tc 1c Tc 2c Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c
185181, 184anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . 24 Nc Clos1 NC NC 2c ↑c Nn Tc 1c Tc 2c Clos1 NC NC 2c ↑c Fin Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c
186178, 185ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . 23 Nc Clos1 NC NC 2c ↑c Nn Tc 1c Tc 2c Clos1 NC NC 2c ↑c Fin Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c
18778, 177, 1863bitri 262 . . . . . . . . . . . . . . . . . . . . . 22 TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC Clos1 NC NC 2c ↑c Fin Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c
18877, 187anbi12i 678 . . . . . . . . . . . . . . . . . . . . 21 SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC Tc Clos1 NC NC 2c ↑c Fin Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c
18967, 188bitri 240 . . . . . . . . . . . . . . . . . . . 20 SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC Tc Clos1 NC NC 2c ↑c Fin Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c
190189exbii 1582 . . . . . . . . . . . . . . . . . . 19 SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC Tc Clos1 NC NC 2c ↑c Fin Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c
191 tcex 6157 . . . . . . . . . . . . . . . . . . . 20 Tc
192 sneq 3744 . . . . . . . . . . . . . . . . . . . . . . 23 Tc Tc
193 clos1eq1 5874 . . . . . . . . . . . . . . . . . . . . . . 23 Tc Clos1 NC NC 2c ↑c Clos1 Tc NC NC 2c ↑c
194192, 193syl 15 . . . . . . . . . . . . . . . . . . . . . 22 Tc Clos1 NC NC 2c ↑c Clos1 Tc NC NC 2c ↑c
195194eleq1d 2419 . . . . . . . . . . . . . . . . . . . . 21 Tc Clos1 NC NC 2c ↑c Fin Clos1 Tc NC NC 2c ↑c Fin
196194nceqd 6110 . . . . . . . . . . . . . . . . . . . . . . 23 Tc Nc Clos1 NC NC 2c ↑c Nc Clos1 Tc NC NC 2c ↑c
197196eqeq1d 2361 . . . . . . . . . . . . . . . . . . . . . 22 Tc Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 1c
198196eqeq1d 2361 . . . . . . . . . . . . . . . . . . . . . 22 Tc Nc Clos1 NC NC 2c ↑c Tc 2c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
199197, 198orbi12d 690 . . . . . . . . . . . . . . . . . . . . 21 Tc Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
200195, 199anbi12d 691 . . . . . . . . . . . . . . . . . . . 20 Tc Clos1 NC NC 2c ↑c Fin Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c Clos1 Tc NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
201191, 200ceqsexv 2894 . . . . . . . . . . . . . . . . . . 19 Tc Clos1 NC NC 2c ↑c Fin Nc Clos1 NC NC 2c ↑c Tc 1c Nc Clos1 NC NC 2c ↑c Tc 2c Clos1 Tc NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
20266, 190, 2013bitri 262 . . . . . . . . . . . . . . . . . 18 SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c Clos1 Tc NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
203202notbii 287 . . . . . . . . . . . . . . . . 17 SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c Clos1 Tc NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
20465, 203anbi12i 678 . . . . . . . . . . . . . . . 16 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c Nc Clos1 NC NC 2c ↑c Clos1 Tc NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
20522, 204bitri 240 . . . . . . . . . . . . . . 15 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c Nc Clos1 NC NC 2c ↑c Clos1 Tc NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Tc 1c Nc Clos1 Tc NC NC 2c ↑c Tc 2c
20621, 205syl6rbbr 255 . . . . . . . . . . . . . 14 NC SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c Nc Spac Spac Tc Fin Nc Spac Tc Tc 1c Nc Spac Tc Tc 2c
207 tceq 6158 . . . . . . . . . . . . . . . . . . . 20 Nc Spac Tc Nc Spac Tc
208207addceq1d 4389 . . . . . . . . . . . . . . . . . . 19 Nc Spac Tc Nc Spac 1c Tc 1c
209208eqeq2d 2364 . . . . . . . . . . . . . . . . . 18 Nc Spac Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc 1c
210207addceq1d 4389 . . . . . . . . . . . . . . . . . . 19 Nc Spac Tc Nc Spac 2c Tc 2c
211210eqeq2d 2364 . . . . . . . . . . . . . . . . . 18 Nc Spac Nc Spac Tc Tc Nc Spac 2c Nc Spac Tc Tc 2c
212209, 211orbi12d 690 . . . . . . . . . . . . . . . . 17 Nc Spac Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c Nc Spac Tc Tc 1c Nc Spac Tc Tc 2c
213212anbi2d 684 . . . . . . . . . . . . . . . 16 Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c Spac Tc Fin Nc Spac Tc Tc 1c Nc Spac Tc Tc 2c
214213notbid 285 . . . . . . . . . . . . . . 15 Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c Spac Tc Fin Nc Spac Tc Tc 1c Nc Spac Tc Tc 2c
215214pm5.32i 618 . . . . . . . . . . . . . 14 Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c Nc Spac Spac Tc Fin Nc Spac Tc Tc 1c Nc Spac Tc Tc 2c
216206, 215syl6bbr 254 . . . . . . . . . . . . 13 NC SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
217216rexbiia 2647 . . . . . . . . . . . 12 NC SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
218 rexanali 2660 . . . . . . . . . . . 12 NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
2197, 217, 2183bitrri 263 . . . . . . . . . . 11 NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC
220219con1bii 321 . . . . . . . . . 10 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
2214, 6, 2203bitri 262 . . . . . . . . 9 ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
222 opelco 4884 . . . . . . . . . . 11 1c AddC AddC 1c
223 brcnv 4892 . . . . . . . . . . . . . 14 AddC AddC
224 brres 4949 . . . . . . . . . . . . . . 15 1c 1c
225 eliniseg 5020 . . . . . . . . . . . . . . . . 17 1c 1c
226225anbi2i 675 . . . . . . . . . . . . . . . 16 1c 1c
227 ancom 437 . . . . . . . . . . . . . . . 16 1c 1c
228226, 227bitri 240 . . . . . . . . . . . . . . 15 1c 1c
229125, 99op1st2nd 5790 . . . . . . . . . . . . . . 15 1c 1c
230224, 228, 2293bitri 262 . . . . . . . . . . . . . 14 1c 1c
231223, 230anbi12i 678 . . . . . . . . . . . . 13 AddC 1c AddC 1c
232 ancom 437 . . . . . . . . . . . . 13 AddC 1c 1c AddC
233231, 232bitri 240 . . . . . . . . . . . 12 AddC 1c 1c AddC
234233exbii 1582 . . . . . . . . . . 11 AddC 1c 1c AddC
235125, 99opex 4588 . . . . . . . . . . . 12 1c
236 breq1 4642 . . . . . . . . . . . 12 1c AddC 1c AddC
237235, 236ceqsexv 2894 . . . . . . . . . . 11 1c AddC 1c AddC
238222, 234, 2373bitri 262 . . . . . . . . . 10 1c AddC 1c AddC
239125, 99braddcfn 5826 . . . . . . . . . 10 1c AddC 1c
240 eqcom 2355 . . . . . . . . . 10 1c 1c
241238, 239, 2403bitri 262 . . . . . . . . 9 1c AddC 1c
242221, 241anbi12i 678 . . . . . . . 8 ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC 1c AddC NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c 1c
243 ancom 437 . . . . . . . 8 NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c 1c 1c NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
244242, 243bitri 240 . . . . . . 7 ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC 1c AddC 1c NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
245244exbii 1582 . . . . . 6 ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC 1c AddC 1c NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
246125, 99addcex 4394 . . . . . . 7 1c
247 eqeq2 2362 . . . . . . . . 9 1c Nc Spac Nc Spac 1c
248247imbi1d 308 . . . . . . . 8 1c Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
249248ralbidv 2634 . . . . . . 7 1c NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
250246, 249ceqsexv 2894 . . . . . 6 1c NC Nc Spac Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
2512, 245, 2503bitri 262 . . . . 5 1c AddC ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
252251abbi2i 2464 . . . 4 1c AddC ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
253252uneq2i 3415 . . 3 <_c We NC 1c AddC ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
254 imor 401 . . . 4 <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
255254abbii 2465 . . 3 <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
2561, 253, 2553eqtr4i 2383 . 2 <_c We NC 1c AddC ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
257 abexv 4324 . . 3 <_c We NC
258 2ndex 5112 . . . . . 6
259 1stex 4739 . . . . . . . 8
260259cnvex 5102 . . . . . . 7
261 snex 4111 . . . . . . 7 1c
262260, 261imaex 4747 . . . . . 6 1c
263258, 262resex 5117 . . . . 5 1c
264 addcfnex 5824 . . . . . 6 AddC
265264cnvex 5102 . . . . 5 AddC
266263, 265coex 4750 . . . 4 1c AddC
267 ssetex 4744 . . . . . . . . . . . . 13 S
268267ins3ex 5798 . . . . . . . . . . . . . . . . . 18 Ins3 S
269267complex 4104 . . . . . . . . . . . . . . . . . . . . . . 23 ∼ S
270269cnvex 5102 . . . . . . . . . . . . . . . . . . . . . 22 ∼ S
271267cnvex 5102 . . . . . . . . . . . . . . . . . . . . . . 23 S
27245imageex 5801 . . . . . . . . . . . . . . . . . . . . . . . . 25 Image NC NC 2c ↑c
273267, 272coex 4750 . . . . . . . . . . . . . . . . . . . . . . . 24 S Image NC NC 2c ↑c
274273fixex 5789 . . . . . . . . . . . . . . . . . . . . . . 23 S Image NC NC 2c ↑c
275271, 274resex 5117 . . . . . . . . . . . . . . . . . . . . . 22 S S Image NC NC 2c ↑c
276270, 275txpex 5785 . . . . . . . . . . . . . . . . . . . . 21 ∼ S S S Image NC NC 2c ↑c
277276rnex 5107 . . . . . . . . . . . . . . . . . . . 20 ∼ S S S Image NC NC 2c ↑c
278277complex 4104 . . . . . . . . . . . . . . . . . . 19 ∼ ∼ S S S Image NC NC 2c ↑c
279278ins2ex 5797 . . . . . . . . . . . . . . . . . 18 Ins2 ∼ ∼ S S S Image NC NC 2c ↑c
280268, 279symdifex 4108 . . . . . . . . . . . . . . . . 17 Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c
281280, 125imaex 4747 . . . . . . . . . . . . . . . 16 Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
282281complex 4104 . . . . . . . . . . . . . . 15 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
283282cnvex 5102 . . . . . . . . . . . . . 14 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
284283siex 4753 . . . . . . . . . . . . 13 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
285267, 284coex 4750 . . . . . . . . . . . 12 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
286285cnvex 5102 . . . . . . . . . . 11 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
287 ncsex 6111 . . . . . . . . . . 11 NC
288286, 287resex 5117 . . . . . . . . . 10 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
289288cnvex 5102 . . . . . . . . 9 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
290289siex 4753 . . . . . . . 8 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
291 tcfnex 6244 . . . . . . . . . . . . 13 TcFn
292291cnvex 5102 . . . . . . . . . . . 12 TcFn
293292siex 4753 . . . . . . . . . . 11 SI TcFn
294293siex 4753 . . . . . . . . . 10 SI SI TcFn
295258cnvex 5102 . . . . . . . . . . . . . . . . . . 19
296295, 261imaex 4747 . . . . . . . . . . . . . . . . . 18 1c
297259, 296resex 5117 . . . . . . . . . . . . . . . . 17 1c
298297cnvex 5102 . . . . . . . . . . . . . . . 16 1c
299264, 298coex 4750 . . . . . . . . . . . . . . 15 AddC 1c
300 snex 4111 . . . . . . . . . . . . . . . . . . 19 2c
301295, 300imaex 4747 . . . . . . . . . . . . . . . . . 18 2c
302259, 301resex 5117 . . . . . . . . . . . . . . . . 17 2c
303302cnvex 5102 . . . . . . . . . . . . . . . 16 2c
304264, 303coex 4750 . . . . . . . . . . . . . . 15 AddC 2c
305299, 304unex 4106 . . . . . . . . . . . . . 14 AddC 1c AddC 2c
306305cnvex 5102 . . . . . . . . . . . . 13 AddC 1c AddC 2c
307292, 306coex 4750 . . . . . . . . . . . 12 TcFn AddC 1c AddC 2c
308 nncex 4396 . . . . . . . . . . . 12 Nn
309307, 308resex 5117 . . . . . . . . . . 11 TcFn AddC 1c AddC 2c Nn
310309, 289coex 4750 . . . . . . . . . 10 TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
311294, 310txpex 5785 . . . . . . . . 9 SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC
312125pw1ex 4303 . . . . . . . . 9 1 1c
313311, 312imaex 4747 . . . . . . . 8 SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c
314290, 313difex 4107 . . . . . . 7 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c
315287pw1ex 4303 . . . . . . . . 9 1 NC
316315pw1ex 4303 . . . . . . . 8 1 1 NC
317316pw1ex 4303 . . . . . . 7 1 1 1 NC
318314, 317imaex 4747 . . . . . 6 SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC
319318complex 4104 . . . . 5 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC
320319uni1ex 4293 . . . 4 ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC
321266, 320imaex 4747 . . 3 1c AddC ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC
322257, 321unex 4106 . 2 <_c We NC 1c AddC ⋃1 ∼ SI S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC SI SI TcFn TcFn AddC 1c AddC 2c Nn S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c NC 1 1c1 1 1 NC
323256, 322eqeltrri 2424 1 <_c We NC NC Nc Spac 1c Spac Tc Fin Nc Spac Tc Tc Nc Spac 1c Nc Spac Tc Tc Nc Spac 2c
Colors of variables: wff setvar class
Syntax hints:   wn 3   wi 4   wo 357   wa 358   w3a 934  wex 1541   wceq 1642   wcel 1710  cab 2339  wral 2614  wrex 2615  cvv 2859   ∼ ccompl 3205   cdif 3206   cun 3207   csymdif 3209  csn 3737  ⋃1cuni1 4133  1cc1c 4134  1 cpw1 4135   Nn cnnc 4373   cplc 4375   Fin cfin 4376  cop 4561  copab 4622   class class class wbr 4639  c1st 4717   S csset 4719   SI csi 4720   ccom 4721  cima 4722  ccnv 4771   crn 4773   cres 4774  cfv 4781  c2nd 4783  (class class class)co 5525   ctxp 5735  cfix 5739   AddC caddcfn 5745   Ins2 cins2 5749   Ins3 cins3 5751  Imagecimage 5753   Clos1 cclos1 5872   We cwe 5895   NC cncs 6088   <_c clec 6089   Nc cnc 6091   Tc ctc 6093  2cc2c 6094   ↑c cce 6096  TcFnctcfn 6097   Spac cspac 6273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-fix 5740  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-fullfun 5768  df-clos1 5873  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-tc 6103  df-2c 6104  df-ce 6106  df-tcfn 6107  df-spac 6274
This theorem is referenced by:  nchoicelem17  6305
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