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Theorem spaccl 6286
Description: Closure law for the special set generator. (Contributed by SF, 13-Mar-2015.)
Assertion
Ref Expression
spaccl NC Spac c 0c NC 2cc Spac

Proof of Theorem spaccl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 956 . . . 4 NC Spac c 0c NC Spac
2 spacval 6282 . . . . 5 NC Spac Clos1 NC NC 2cc
323ad2ant1 976 . . . 4 NC Spac c 0c NC Spac Clos1 NC NC 2cc
41, 3eleqtrd 2429 . . 3 NC Spac c 0c NC Clos1 NC NC 2cc
5 spacssnc 6284 . . . . . 6 NC Spac NC
65sselda 3273 . . . . 5 NC Spac NC
763adant3 975 . . . 4 NC Spac c 0c NC NC
8 simp3 957 . . . . 5 NC Spac c 0c NC c 0c NC
9 2nnc 6167 . . . . . 6 2c Nn
10 ceclnn1 6189 . . . . . 6 2c Nn NC c 0c NC 2cc NC
119, 10mp3an1 1264 . . . . 5 NC c 0c NC 2cc NC
127, 8, 11syl2anc 642 . . . 4 NC Spac c 0c NC 2cc NC
13 eqidd 2354 . . . 4 NC Spac c 0c NC 2cc 2cc
14 ovex 5551 . . . . . 6 2cc
15 eleq1 2413 . . . . . . . 8 NC NC
16 oveq2 5531 . . . . . . . . 9 2cc 2cc
1716eqeq2d 2364 . . . . . . . 8 2cc 2cc
1815, 173anbi13d 1254 . . . . . . 7 NC NC 2cc NC NC 2cc
19 eleq1 2413 . . . . . . . 8 2cc NC 2cc NC
20 eqeq1 2359 . . . . . . . 8 2cc 2cc 2cc 2cc
2119, 203anbi23d 1255 . . . . . . 7 2cc NC NC 2cc NC 2cc NC 2cc 2cc
22 eqid 2353 . . . . . . 7 NC NC 2cc NC NC 2cc
2318, 21, 22brabg 4706 . . . . . 6 Spac 2cc NC NC 2cc 2cc NC 2cc NC 2cc 2cc
2414, 23mpan2 652 . . . . 5 Spac NC NC 2cc 2cc NC 2cc NC 2cc 2cc
25243ad2ant2 977 . . . 4 NC Spac c 0c NC NC NC 2cc 2cc NC 2cc NC 2cc 2cc
267, 12, 13, 25mpbir3and 1135 . . 3 NC Spac c 0c NC NC NC 2cc 2cc
27 eqid 2353 . . . 4 Clos1 NC NC 2cc Clos1 NC NC 2cc
2827clos1conn 5879 . . 3 Clos1 NC NC 2cc NC NC 2cc 2cc 2cc Clos1 NC NC 2cc
294, 26, 28syl2anc 642 . 2 NC Spac c 0c NC 2cc Clos1 NC NC 2cc
3029, 3eleqtrrd 2430 1 NC Spac c 0c NC 2cc Spac
Colors of variables: wff setvar class
Syntax hints:   wi 4   wb 176   w3a 934   wceq 1642   wcel 1710  cvv 2859  csn 3737   Nn cnnc 4373  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-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-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:  nchoicelem6  6294
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