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Theorem nchoicelem11 6297
Description: Lemma for nchoice 6306. Set up stratification for nchoicelem12 6298. (Contributed by SF, 18-Mar-2015.)
Assertion
Ref Expression
nchoicelem11 NC Nc Spac Tc Nc Spac Nn
Distinct variable group:   ,
Proof of Theorem nchoicelem11
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . 5
21elcompl 3225 . . . 4 ∼ NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC
3 elimapw13 4946 . . . . . . 7 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC NC NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
4 tccl 6160 . . . . . . . . . . . . . 14 NC Tc NC
5 spacval 6280 . . . . . . . . . . . . . 14 Tc NC Spac Tc Clos1 Tc NC NC 2c ↑c
64, 5syl 15 . . . . . . . . . . . . 13 NC Spac Tc Clos1 Tc NC NC 2c ↑c
76nceqd 6110 . . . . . . . . . . . 12 NC Nc Spac Tc Nc Clos1 Tc NC NC 2c ↑c
87eqeq2d 2364 . . . . . . . . . . 11 NC Nc Spac Tc Nc Clos1 Tc NC NC 2c ↑c
9 finnc 6242 . . . . . . . . . . . 12 Spac Fin Nc Spac Nn
10 spacval 6280 . . . . . . . . . . . . 13 NC Spac Clos1 NC NC 2c ↑c
1110eleq1d 2419 . . . . . . . . . . . 12 NC Spac Fin Clos1 NC NC 2c ↑c Fin
129, 11syl5bbr 250 . . . . . . . . . . 11 NC Nc Spac Nn Clos1 NC NC 2c ↑c Fin
138, 12imbi12d 311 . . . . . . . . . 10 NC Nc Spac Tc Nc Spac Nn Nc Clos1 Tc NC NC 2c ↑c Clos1 NC NC 2c ↑c Fin
1413notbid 285 . . . . . . . . 9 NC Nc Spac Tc Nc Spac Nn Nc Clos1 Tc NC NC 2c ↑c Clos1 NC NC 2c ↑c Fin
15 eldif 3221 . . . . . . . . . 10 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
16 opelco 4884 . . . . . . . . . . . . . 14 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn SI SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
17 snex 4111 . . . . . . . . . . . . . . . . . . 19
1817brsnsi1 5775 . . . . . . . . . . . . . . . . . 18 SI SI TcFn SI TcFn
1918anbi1i 676 . . . . . . . . . . . . . . . . 17 SI SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
20 19.41v 1901 . . . . . . . . . . . . . . . . . 18 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
2120bicomi 193 . . . . . . . . . . . . . . . . 17 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
2219, 21bitri 240 . . . . . . . . . . . . . . . 16 SI SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
2322exbii 1582 . . . . . . . . . . . . . . 15 SI SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
24 excom 1741 . . . . . . . . . . . . . . . 16 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
25 anass 630 . . . . . . . . . . . . . . . . . . . 20 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
2625exbii 1582 . . . . . . . . . . . . . . . . . . 19 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
27 snex 4111 . . . . . . . . . . . . . . . . . . . 20
28 breq1 4642 . . . . . . . . . . . . . . . . . . . . 21 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
2928anbi2d 684 . . . . . . . . . . . . . . . . . . . 20 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
3027, 29ceqsexv 2894 . . . . . . . . . . . . . . . . . . 19 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
3126, 30bitri 240 . . . . . . . . . . . . . . . . . 18 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
3231exbii 1582 . . . . . . . . . . . . . . . . 17 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
33 snex 4111 . . . . . . . . . . . . . . . . . . . . . 22
3433brsnsi1 5775 . . . . . . . . . . . . . . . . . . . . 21 SI TcFn TcFn
3534anbi1i 676 . . . . . . . . . . . . . . . . . . . 20 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
36 19.41v 1901 . . . . . . . . . . . . . . . . . . . 20 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
3735, 36bitr4i 243 . . . . . . . . . . . . . . . . . . 19 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
3837exbii 1582 . . . . . . . . . . . . . . . . . 18 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
39 excom 1741 . . . . . . . . . . . . . . . . . . 19 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
40 anass 630 . . . . . . . . . . . . . . . . . . . . . 22 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
4140exbii 1582 . . . . . . . . . . . . . . . . . . . . 21 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
42 snex 4111 . . . . . . . . . . . . . . . . . . . . . 22
43 sneq 3744 . . . . . . . . . . . . . . . . . . . . . . . 24
4443breq1d 4649 . . . . . . . . . . . . . . . . . . . . . . 23 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
4544anbi2d 684 . . . . . . . . . . . . . . . . . . . . . 22 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
4642, 45ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . 21 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
4741, 46bitri 240 . . . . . . . . . . . . . . . . . . . 20 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
4847exbii 1582 . . . . . . . . . . . . . . . . . . 19 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
4939, 48bitri 240 . . . . . . . . . . . . . . . . . 18 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
5038, 49bitri 240 . . . . . . . . . . . . . . . . 17 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
5132, 50bitri 240 . . . . . . . . . . . . . . . 16 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
5224, 51bitri 240 . . . . . . . . . . . . . . 15 SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
5323, 52bitri 240 . . . . . . . . . . . . . 14 SI SI TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
5416, 53bitri 240 . . . . . . . . . . . . 13 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
55 vex 2862 . . . . . . . . . . . . . . . 16
5655brtcfn 6245 . . . . . . . . . . . . . . 15 TcFn Tc
57 df-br 4640 . . . . . . . . . . . . . . . . 17 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
58 opelcnv 4893 . . . . . . . . . . . . . . . . . 18 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
59 elin 3219 . . . . . . . . . . . . . . . . . . 19 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
60 snex 4111 . . . . . . . . . . . . . . . . . . . . 21
61 opelxp 4811 . . . . . . . . . . . . . . . . . . . . 21 NC NC
6260, 61mpbiran2 885 . . . . . . . . . . . . . . . . . . . 20 NC NC
63 ancom 437 . . . . . . . . . . . . . . . . . . . . . . . . . 26 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
6442brsnsi2 5776 . . . . . . . . . . . . . . . . . . . . . . . . . . 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
6564anbi1i 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
6663, 65bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . 25 S 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 S
67 19.41v 1901 . . . . . . . . . . . . . . . . . . . . . . . . 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
6866, 67bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . 24 S 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 S
6968exbii 1582 . . . . . . . . . . . . . . . . . . . . . . 23 S 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 S
70 excom 1741 . . . . . . . . . . . . . . . . . . . . . . 23 ∼ 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
7169, 70bitri 240 . . . . . . . . . . . . . . . . . . . . . 22 S 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 S
72 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
7372exbii 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
74 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . 26
75 breq2 4643 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 S S
7675anbi2d 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
7774, 76ceqsexv 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
7873, 77bitri 240 . . . . . . . . . . . . . . . . . . . . . . . 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
79 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
80 spacvallem1 6279 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 NC NC 2c ↑c
8180, 42nchoicelem10 6296 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
8279, 81bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . 25 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
83 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . 26 S S
84 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8584, 1brssetsn 4759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 S
8683, 85bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . 25 S
8782, 86anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . 24 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
8878, 87bitri 240 . . . . . . . . . . . . . . . . . . . . . . 23 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
8988exbii 1582 . . . . . . . . . . . . . . . . . . . . . 22 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
9071, 89bitri 240 . . . . . . . . . . . . . . . . . . . . 21 S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
91 opelco 4884 . . . . . . . . . . . . . . . . . . . . 21 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S S SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
92 df-clel 2349 . . . . . . . . . . . . . . . . . . . . 21 Clos1 NC NC 2c ↑c Clos1 NC NC 2c ↑c
9390, 91, 923bitr4i 268 . . . . . . . . . . . . . . . . . . . 20 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Clos1 NC NC 2c ↑c
9462, 93anbi12i 678 . . . . . . . . . . . . . . . . . . 19 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S NC Clos1 NC NC 2c ↑c
9559, 94bitri 240 . . . . . . . . . . . . . . . . . 18 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S NC Clos1 NC NC 2c ↑c
9658, 95bitri 240 . . . . . . . . . . . . . . . . 17 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S NC Clos1 NC NC 2c ↑c
9757, 96bitri 240 . . . . . . . . . . . . . . . 16 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S NC Clos1 NC NC 2c ↑c
9842, 80clos1ex 5876 . . . . . . . . . . . . . . . . 17 Clos1 NC NC 2c ↑c
9998eqnc2 6129 . . . . . . . . . . . . . . . 16 Nc Clos1 NC NC 2c ↑c NC Clos1 NC NC 2c ↑c
10097, 99bitr4i 243 . . . . . . . . . . . . . . 15 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Nc Clos1 NC NC 2c ↑c
10156, 100anbi12i 678 . . . . . . . . . . . . . 14 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Tc Nc Clos1 NC NC 2c ↑c
102101exbii 1582 . . . . . . . . . . . . 13 TcFn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S Tc Nc Clos1 NC NC 2c ↑c
10354, 102bitri 240 . . . . . . . . . . . 12 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn Tc Nc Clos1 NC NC 2c ↑c
104 tcex 6157 . . . . . . . . . . . . 13 Tc
105 sneq 3744 . . . . . . . . . . . . . . . 16 Tc Tc
106 clos1eq1 5874 . . . . . . . . . . . . . . . 16 Tc Clos1 NC NC 2c ↑c Clos1 Tc NC NC 2c ↑c
107105, 106syl 15 . . . . . . . . . . . . . . 15 Tc Clos1 NC NC 2c ↑c Clos1 Tc NC NC 2c ↑c
108107nceqd 6110 . . . . . . . . . . . . . 14 Tc Nc Clos1 NC NC 2c ↑c Nc Clos1 Tc NC NC 2c ↑c
109108eqeq2d 2364 . . . . . . . . . . . . 13 Tc Nc Clos1 NC NC 2c ↑c Nc Clos1 Tc NC NC 2c ↑c
110104, 109ceqsexv 2894 . . . . . . . . . . . 12 Tc Nc Clos1 NC NC 2c ↑c Nc Clos1 Tc NC NC 2c ↑c
111103, 110bitri 240 . . . . . . . . . . 11 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn Nc Clos1 Tc NC NC 2c ↑c
112 df-br 4640 . . . . . . . . . . . . . . 15 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
11380, 33nchoicelem10 6296 . . . . . . . . . . . . . . 15 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
114112, 113bitri 240 . . . . . . . . . . . . . 14 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Clos1 NC NC 2c ↑c
115114rexbii 2639 . . . . . . . . . . . . 13 Fin ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Clos1 NC NC 2c ↑c
116 opelxp 4811 . . . . . . . . . . . . . . 15 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
1171, 116mpbiran2 885 . . . . . . . . . . . . . 14 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
118 snelpw1 4146 . . . . . . . . . . . . . . 15 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
119 snelpw1 4146 . . . . . . . . . . . . . . . 16 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
120 elima 4754 . . . . . . . . . . . . . . . 16 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Fin ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
121119, 120bitri 240 . . . . . . . . . . . . . . 15 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Fin ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
122118, 121bitri 240 . . . . . . . . . . . . . 14 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Fin ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
123117, 122bitri 240 . . . . . . . . . . . . 13 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Fin ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
124 risset 2661 . . . . . . . . . . . . 13 Clos1 NC NC 2c ↑c Fin Fin Clos1 NC NC 2c ↑c
125115, 123, 1243bitr4i 268 . . . . . . . . . . . 12 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Clos1 NC NC 2c ↑c Fin
126125notbii 287 . . . . . . . . . . 11 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Clos1 NC NC 2c ↑c Fin
127111, 126anbi12i 678 . . . . . . . . . 10 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Nc Clos1 Tc NC NC 2c ↑c Clos1 NC NC 2c ↑c Fin
128 annim 414 . . . . . . . . . 10 Nc Clos1 Tc NC NC 2c ↑c Clos1 NC NC 2c ↑c Fin Nc Clos1 Tc NC NC 2c ↑c Clos1 NC NC 2c ↑c Fin
12915, 127, 1283bitri 262 . . . . . . . . 9 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Nc Clos1 Tc NC NC 2c ↑c Clos1 NC NC 2c ↑c Fin
13014, 129syl6rbbr 255 . . . . . . . 8 NC NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin Nc Spac Tc Nc Spac Nn
131130rexbiia 2647 . . . . . . 7 NC NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin NC Nc Spac Tc Nc Spac Nn
1323, 131bitri 240 . . . . . 6 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC NC Nc Spac Tc Nc Spac Nn
133 rexnal 2625 . . . . . 6 NC Nc Spac Tc Nc Spac Nn NC Nc Spac Tc Nc Spac Nn
134132, 133bitr2i 241 . . . . 5 NC Nc Spac Tc Nc Spac Nn NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC
135134con1bii 321 . . . 4 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC NC Nc Spac Tc Nc Spac Nn
1362, 135bitri 240 . . 3 ∼ NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC NC Nc Spac Tc Nc Spac Nn
137136abbi2i 2464 . 2 ∼ NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC NC Nc Spac Tc Nc Spac Nn
138 ncsex 6111 . . . . . . . . 9 NC
139 vvex 4109 . . . . . . . . 9
140138, 139xpex 5115 . . . . . . . 8 NC
141 ssetex 4744 . . . . . . . . . . . . . 14 S
142141ins3ex 5798 . . . . . . . . . . . . 13 Ins3 S
143141complex 4104 . . . . . . . . . . . . . . . . . 18 ∼ S
144143cnvex 5102 . . . . . . . . . . . . . . . . 17 ∼ S
145141cnvex 5102 . . . . . . . . . . . . . . . . . 18 S
14680imageex 5801 . . . . . . . . . . . . . . . . . . . 20 Image NC NC 2c ↑c
147141, 146coex 4750 . . . . . . . . . . . . . . . . . . 19 S Image NC NC 2c ↑c
148147fixex 5789 . . . . . . . . . . . . . . . . . 18 S Image NC NC 2c ↑c
149145, 148resex 5117 . . . . . . . . . . . . . . . . 17 S S Image NC NC 2c ↑c
150144, 149txpex 5785 . . . . . . . . . . . . . . . 16 ∼ S S S Image NC NC 2c ↑c
151150rnex 5107 . . . . . . . . . . . . . . 15 ∼ S S S Image NC NC 2c ↑c
152151complex 4104 . . . . . . . . . . . . . 14 ∼ ∼ S S S Image NC NC 2c ↑c
153152ins2ex 5797 . . . . . . . . . . . . 13 Ins2 ∼ ∼ S S S Image NC NC 2c ↑c
154142, 153symdifex 4108 . . . . . . . . . . . 12 Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c
155 1cex 4142 . . . . . . . . . . . 12 1c
156154, 155imaex 4747 . . . . . . . . . . 11 Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
157156complex 4104 . . . . . . . . . 10 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
158157siex 4753 . . . . . . . . 9 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c
159158, 145coex 4750 . . . . . . . 8 SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
160140, 159inex 4105 . . . . . . 7 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
161160cnvex 5102 . . . . . 6 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S
162 tcfnex 6243 . . . . . . . 8 TcFn
163162siex 4753 . . . . . . 7 SI TcFn
164163siex 4753 . . . . . 6 SI SI TcFn
165161, 164coex 4750 . . . . 5 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn
166 finex 4397 . . . . . . . . 9 Fin
167157, 166imaex 4747 . . . . . . . 8 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
168167pw1ex 4303 . . . . . . 7 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
169168pw1ex 4303 . . . . . 6 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
170169, 139xpex 5115 . . . . 5 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
171165, 170difex 4107 . . . 4 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin
172138pw1ex 4303 . . . . . 6 1 NC
173172pw1ex 4303 . . . . 5 1 1 NC
174173pw1ex 4303 . . . 4 1 1 1 NC
175171, 174imaex 4747 . . 3 NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC
176175complex 4104 . 2 ∼ NC SI ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c S SI SI TcFn 1 1 ∼ Ins3 S Ins2 ∼ ∼ S S S Image NC NC 2c ↑c 1c Fin 1 1 1 NC
177137, 176eqeltrri 2424 1 NC Nc Spac Tc Nc Spac Nn
Colors of variables: wff setvar class
Syntax hints:   wn 3   wi 4   wa 358   w3a 934  wex 1541   wceq 1642   wcel 1710  cab 2339  wral 2614  wrex 2615  cvv 2859   ∼ ccompl 3205   cdif 3206   cin 3208   csymdif 3209  csn 3737  1cc1c 4134  1 cpw1 4135   Nn cnnc 4373   Fin cfin 4376  cop 4561  copab 4622   class class class wbr 4639   S csset 4719   SI csi 4720   ccom 4721  cima 4722   cxp 4770  ccnv 4771   crn 4773   cres 4774  cfv 4781  (class class class)co 5525   ctxp 5735  cfix 5739   Ins2 cins2 5749   Ins3 cins3 5751  Imagecimage 5753   Clos1 cclos1 5872   NC cncs 6088   Nc cnc 6091   Tc ctc 6093  2cc2c 6094   ↑c cce 6096  TcFnctcfn 6097   Spac cspac 6271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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-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-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 6272
This theorem is referenced by:  nchoicelem12  6298
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