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Theorem nchoicelem6 6294
 Description: Lemma for nchoice 6308. Split the special set generator into base and inductive values. Theorem 6.6 of [Specker] p. 973. (Contributed by SF, 13-Mar-2015.)
Assertion
Ref Expression
nchoicelem6 NC c 0c NC Spac Spac 2cc

Proof of Theorem nchoicelem6
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3 NC c 0c NC NC
2 snex 4111 . . . . 5
3 fvex 5339 . . . . 5 Spac 2cc
42, 3unex 4106 . . . 4 Spac 2cc
54a1i 10 . . 3 NC c 0c NC Spac 2cc
6 snidg 3758 . . . . 5 NC
76adantr 451 . . . 4 NC c 0c NC
8 elun1 3430 . . . 4 Spac 2cc
97, 8syl 15 . . 3 NC c 0c NC Spac 2cc
10 elun 3220 . . . . . . . . . 10 Spac 2cc Spac 2cc
11 elsn 3748 . . . . . . . . . . 11
1211orbi1i 506 . . . . . . . . . 10 Spac 2cc Spac 2cc
1310, 12bitri 240 . . . . . . . . 9 Spac 2cc Spac 2cc
14 spacssnc 6284 . . . . . . . . . . . . . . 15 NC Spac NC
1514adantr 451 . . . . . . . . . . . . . 14 NC c 0c NC Spac NC
16 spacid 6285 . . . . . . . . . . . . . . . 16 NC Spac
1716adantr 451 . . . . . . . . . . . . . . 15 NC c 0c NC Spac
18 simpr 447 . . . . . . . . . . . . . . 15 NC c 0c NC c 0c NC
19 spaccl 6286 . . . . . . . . . . . . . . 15 NC Spac c 0c NC 2cc Spac
201, 17, 18, 19syl3anc 1182 . . . . . . . . . . . . . 14 NC c 0c NC 2cc Spac
2115, 20sseldd 3274 . . . . . . . . . . . . 13 NC c 0c NC 2cc NC
22 spacid 6285 . . . . . . . . . . . . 13 2cc NC 2cc Spac 2cc
2321, 22syl 15 . . . . . . . . . . . 12 NC c 0c NC 2cc Spac 2cc
24 oveq2 5531 . . . . . . . . . . . . 13 2cc 2cc
2524eleq1d 2419 . . . . . . . . . . . 12 2cc Spac 2cc 2cc Spac 2cc
2623, 25syl5ibrcom 213 . . . . . . . . . . 11 NC c 0c NC 2cc Spac 2cc
2726adantr 451 . . . . . . . . . 10 NC c 0c NC c 0c NC 2cc Spac 2cc
28 2nnc 6167 . . . . . . . . . . . . . 14 2c Nn
29 ceclnn1 6189 . . . . . . . . . . . . . 14 2c Nn NC c 0c NC 2cc NC
3028, 29mp3an1 1264 . . . . . . . . . . . . 13 NC c 0c NC 2cc NC
3130adantr 451 . . . . . . . . . . . 12 NC c 0c NC c 0c NC Spac 2cc 2cc NC
32 simprr 733 . . . . . . . . . . . 12 NC c 0c NC c 0c NC Spac 2cc Spac 2cc
33 simprl 732 . . . . . . . . . . . 12 NC c 0c NC c 0c NC Spac 2cc c 0c NC
34 spaccl 6286 . . . . . . . . . . . 12 2cc NC Spac 2cc c 0c NC 2cc Spac 2cc
3531, 32, 33, 34syl3anc 1182 . . . . . . . . . . 11 NC c 0c NC c 0c NC Spac 2cc 2cc Spac 2cc
3635expr 598 . . . . . . . . . 10 NC c 0c NC c 0c NC Spac 2cc 2cc Spac 2cc
3727, 36jaod 369 . . . . . . . . 9 NC c 0c NC c 0c NC Spac 2cc 2cc Spac 2cc
3813, 37syl5bi 208 . . . . . . . 8 NC c 0c NC c 0c NC Spac 2cc 2cc Spac 2cc
3938ex 423 . . . . . . 7 NC c 0c NC c 0c NC Spac 2cc 2cc Spac 2cc
4039com23 72 . . . . . 6 NC c 0c NC Spac 2cc c 0c NC 2cc Spac 2cc
4140imp3a 420 . . . . 5 NC c 0c NC Spac 2cc c 0c NC 2cc Spac 2cc
42 elun2 3431 . . . . 5 2cc Spac 2cc 2cc Spac 2cc
4341, 42syl6 29 . . . 4 NC c 0c NC Spac 2cc c 0c NC 2cc Spac 2cc
4443ralrimivw 2698 . . 3 NC c 0c NC Spac Spac 2cc c 0c NC 2cc Spac 2cc
45 spacind 6287 . . 3 NC Spac 2cc Spac 2cc Spac Spac 2cc c 0c NC 2cc Spac 2cc Spac Spac 2cc
461, 5, 9, 44, 45syl22anc 1183 . 2 NC c 0c NC Spac Spac 2cc
4716snssd 3853 . . . 4 NC Spac
4847adantr 451 . . 3 NC c 0c NC Spac
49 fvex 5339 . . . . 5 Spac
5049a1i 10 . . . 4 NC c 0c NC Spac
51 spaccl 6286 . . . . . . 7 NC Spac c 0c NC 2cc Spac
52513expib 1154 . . . . . 6 NC Spac c 0c NC 2cc Spac
5352adantr 451 . . . . 5 NC c 0c NC Spac c 0c NC 2cc Spac
5453ralrimivw 2698 . . . 4 NC c 0c NC Spac 2cc Spac c 0c NC 2cc Spac
55 spacind 6287 . . . 4 2cc NC Spac 2cc Spac Spac 2cc Spac c 0c NC 2cc Spac Spac 2cc Spac
5621, 50, 20, 54, 55syl22anc 1183 . . 3 NC c 0c NC Spac 2cc Spac
5748, 56unssd 3439 . 2 NC c 0c NC Spac 2cc Spac
5846, 57eqssd 3289 1 NC c 0c NC Spac Spac 2cc
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 357   wa 358   wceq 1642   wcel 1710  wral 2614  cvv 2859   cun 3207   wss 3257  csn 3737   Nn cnnc 4373  0cc0c 4374  cfv 4781  (class class class)co 5525   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:  nchoicelem7  6295  nchoicelem14  6302
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