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Theorem spacind 6287
 Description: Inductive law for the special set generator. (Contributed by SF, 13-Mar-2015.)
Assertion
Ref Expression
spacind NC Spac c 0c NC 2cc Spac
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem spacind
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2867 . 2
2 spacval 6282 . . . . 5 NC Spac Clos1 NC NC 2cc
32adantr 451 . . . 4 NC Spac Clos1 NC NC 2cc
43adantr 451 . . 3 NC Spac c 0c NC 2cc Spac Clos1 NC NC 2cc
5 simplr 731 . . . 4 NC Spac c 0c NC 2cc
6 snssi 3852 . . . . . 6
76adantr 451 . . . . 5 Spac c 0c NC 2cc
87adantl 452 . . . 4 NC Spac c 0c NC 2cc
9 spacssnc 6284 . . . . . . . . . . 11 NC Spac NC
109sseld 3272 . . . . . . . . . 10 NC Spac NC
11 2nc 6168 . . . . . . . . . . . . . . . . . . . 20 2c NC
12 ceclr 6187 . . . . . . . . . . . . . . . . . . . . 21 2c NC NC 2cc NC 2cc 0c NC c 0c NC
1312simprd 449 . . . . . . . . . . . . . . . . . . . 20 2c NC NC 2cc NC c 0c NC
1411, 13mp3an1 1264 . . . . . . . . . . . . . . . . . . 19 NC 2cc NC c 0c NC
1514ex 423 . . . . . . . . . . . . . . . . . 18 NC 2cc NC c 0c NC
1615imim1d 69 . . . . . . . . . . . . . . . . 17 NC c 0c NC 2cc 2cc NC 2cc
1716a1dd 42 . . . . . . . . . . . . . . . 16 NC c 0c NC 2cc NC 2cc NC 2cc
1817adantl 452 . . . . . . . . . . . . . . 15 NC NC c 0c NC 2cc NC 2cc NC 2cc
19 3anass 938 . . . . . . . . . . . . . . . . . . 19 NC NC 2cc NC NC 2cc
2019imbi1i 315 . . . . . . . . . . . . . . . . . 18 NC NC 2cc NC NC 2cc
21 impexp 433 . . . . . . . . . . . . . . . . . 18 NC NC 2cc NC NC 2cc
2220, 21bitri 240 . . . . . . . . . . . . . . . . 17 NC NC 2cc NC NC 2cc
2322albii 1566 . . . . . . . . . . . . . . . 16 NC NC 2cc NC NC 2cc
24 19.21v 1890 . . . . . . . . . . . . . . . . 17 NC NC 2cc NC NC 2cc
25 impexp 433 . . . . . . . . . . . . . . . . . . . . 21 NC 2cc NC 2cc
26 bi2.04 350 . . . . . . . . . . . . . . . . . . . . 21 NC 2cc 2cc NC
2725, 26bitri 240 . . . . . . . . . . . . . . . . . . . 20 NC 2cc 2cc NC
2827albii 1566 . . . . . . . . . . . . . . . . . . 19 NC 2cc 2cc NC
29 ovex 5551 . . . . . . . . . . . . . . . . . . . 20 2cc
30 eleq1 2413 . . . . . . . . . . . . . . . . . . . . 21 2cc NC 2cc NC
31 eleq1 2413 . . . . . . . . . . . . . . . . . . . . 21 2cc 2cc
3230, 31imbi12d 311 . . . . . . . . . . . . . . . . . . . 20 2cc NC 2cc NC 2cc
3329, 32ceqsalv 2885 . . . . . . . . . . . . . . . . . . 19 2cc NC 2cc NC 2cc
3428, 33bitri 240 . . . . . . . . . . . . . . . . . 18 NC 2cc 2cc NC 2cc
3534imbi2i 303 . . . . . . . . . . . . . . . . 17 NC NC 2cc NC 2cc NC 2cc
3624, 35bitri 240 . . . . . . . . . . . . . . . 16 NC NC 2cc NC 2cc NC 2cc
3723, 36bitri 240 . . . . . . . . . . . . . . 15 NC NC 2cc NC 2cc NC 2cc
3818, 37syl6ibr 218 . . . . . . . . . . . . . 14 NC NC c 0c NC 2cc NC NC 2cc
39 vex 2862 . . . . . . . . . . . . . . . . 17
40 vex 2862 . . . . . . . . . . . . . . . . 17
41 eleq1 2413 . . . . . . . . . . . . . . . . . 18 NC NC
42 oveq2 5531 . . . . . . . . . . . . . . . . . . 19 2cc 2cc
4342eqeq2d 2364 . . . . . . . . . . . . . . . . . 18 2cc 2cc
4441, 433anbi13d 1254 . . . . . . . . . . . . . . . . 17 NC NC 2cc NC NC 2cc
45 eleq1 2413 . . . . . . . . . . . . . . . . . 18 NC NC
46 eqeq1 2359 . . . . . . . . . . . . . . . . . 18 2cc 2cc
4745, 463anbi23d 1255 . . . . . . . . . . . . . . . . 17 NC NC 2cc NC NC 2cc
48 eqid 2353 . . . . . . . . . . . . . . . . 17 NC NC 2cc NC NC 2cc
4939, 40, 44, 47, 48brab 4709 . . . . . . . . . . . . . . . 16 NC NC 2cc NC NC 2cc
5049imbi1i 315 . . . . . . . . . . . . . . 15 NC NC 2cc NC NC 2cc
5150albii 1566 . . . . . . . . . . . . . 14 NC NC 2cc NC NC 2cc
5238, 51syl6ibr 218 . . . . . . . . . . . . 13 NC NC c 0c NC 2cc NC NC 2cc
5352imim2d 48 . . . . . . . . . . . 12 NC NC c 0c NC 2cc NC NC 2cc
54 impexp 433 . . . . . . . . . . . 12 c 0c NC 2cc c 0c NC 2cc
55 impexp 433 . . . . . . . . . . . . . 14 NC NC 2cc NC NC 2cc
5655albii 1566 . . . . . . . . . . . . 13 NC NC 2cc NC NC 2cc
57 19.21v 1890 . . . . . . . . . . . . 13 NC NC 2cc NC NC 2cc
5856, 57bitri 240 . . . . . . . . . . . 12 NC NC 2cc NC NC 2cc
5953, 54, 583imtr4g 261 . . . . . . . . . . 11 NC NC c 0c NC 2cc NC NC 2cc
6059ex 423 . . . . . . . . . 10 NC NC c 0c NC 2cc NC NC 2cc
6110, 60syld 40 . . . . . . . . 9 NC Spac c 0c NC 2cc NC NC 2cc
6261imp 418 . . . . . . . 8 NC Spac c 0c NC 2cc NC NC 2cc
6362ralimdva 2692 . . . . . . 7 NC Spac c 0c NC 2cc Spac NC NC 2cc
64 raleq 2807 . . . . . . . 8 Spac Clos1 NC NC 2cc Spac NC NC 2cc Clos1 NC NC 2cc NC NC 2cc
652, 64syl 15 . . . . . . 7 NC Spac NC NC 2cc Clos1 NC NC 2cc NC NC 2cc
6663, 65sylibd 205 . . . . . 6 NC Spac c 0c NC 2cc Clos1 NC NC 2cc NC NC 2cc
6766imp 418 . . . . 5 NC Spac c 0c NC 2cc Clos1 NC NC 2cc NC NC 2cc
6867ad2ant2rl 729 . . . 4 NC Spac c 0c NC 2cc Clos1 NC NC 2cc NC NC 2cc
69 snex 4111 . . . . 5
70 spacvallem1 6281 . . . . 5 NC NC 2cc
71 eqid 2353 . . . . 5 Clos1 NC NC 2cc Clos1 NC NC 2cc
7269, 70, 71clos1induct 5880 . . . 4 Clos1 NC NC 2cc NC NC 2cc Clos1 NC NC 2cc
735, 8, 68, 72syl3anc 1182 . . 3 NC Spac c 0c NC 2cc Clos1 NC NC 2cc
744, 73eqsstrd 3305 . 2 NC Spac c 0c NC 2cc Spac
751, 74sylanl2 632 1 NC Spac c 0c NC 2cc Spac
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   w3a 934  wal 1540   wceq 1642   wcel 1710  wral 2614  cvv 2859   wss 3257  csn 3737  0cc0c 4374  copab 4622   class class class wbr 4639  cfv 4781  (class class class)co 5525   Clos1 cclos1 5872   NC cncs 6088  2cc2c 6094   ↑c cce 6096   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-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-compose 5748  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-2c 6104  df-ce 6106  df-spac 6274 This theorem is referenced by:  spacis  6288  nchoicelem6  6294
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