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Theorem nchoicelem12 6300
 Description: Lemma for nchoice 6308. If the T-raising of a cardinal yields a finite special set, then so does the initial set. Theorem 7.1 of [Specker] p. 974. (Contributed by SF, 18-Mar-2015.)
Assertion
Ref Expression
nchoicelem12 NC Spac Tc Fin Spac Fin

Proof of Theorem nchoicelem12
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 finnc 6243 . . . 4 Spac Tc Fin Nc Spac Tc Nn
2 risset 2661 . . . 4 Nc Spac Tc Nn Nn Nc Spac Tc
31, 2bitri 240 . . 3 Spac Tc Fin Nn Nc Spac Tc
4 nchoicelem11 6299 . . . . . . 7 NC Nc Spac Tc Nc Spac Nn
5 eqeq1 2359 . . . . . . . . 9 0c Nc Spac Tc 0c Nc Spac Tc
65imbi1d 308 . . . . . . . 8 0c Nc Spac Tc Nc Spac Nn 0c Nc Spac Tc Nc Spac Nn
76ralbidv 2634 . . . . . . 7 0c NC Nc Spac Tc Nc Spac Nn NC 0c Nc Spac Tc Nc Spac Nn
8 eqeq1 2359 . . . . . . . . 9 Nc Spac Tc Nc Spac Tc
98imbi1d 308 . . . . . . . 8 Nc Spac Tc Nc Spac Nn Nc Spac Tc Nc Spac Nn
109ralbidv 2634 . . . . . . 7 NC Nc Spac Tc Nc Spac Nn NC Nc Spac Tc Nc Spac Nn
11 eqeq1 2359 . . . . . . . . . 10 1c Nc Spac Tc 1c Nc Spac Tc
1211imbi1d 308 . . . . . . . . 9 1c Nc Spac Tc Nc Spac Nn 1c Nc Spac Tc Nc Spac Nn
1312ralbidv 2634 . . . . . . . 8 1c NC Nc Spac Tc Nc Spac Nn NC 1c Nc Spac Tc Nc Spac Nn
14 tceq 6158 . . . . . . . . . . . . 13 Tc Tc
1514fveq2d 5332 . . . . . . . . . . . 12 Spac Tc Spac Tc
1615nceqd 6110 . . . . . . . . . . 11 Nc Spac Tc Nc Spac Tc
1716eqeq2d 2364 . . . . . . . . . 10 1c Nc Spac Tc 1c Nc Spac Tc
18 fveq2 5328 . . . . . . . . . . . 12 Spac Spac
1918nceqd 6110 . . . . . . . . . . 11 Nc Spac Nc Spac
2019eleq1d 2419 . . . . . . . . . 10 Nc Spac Nn Nc Spac Nn
2117, 20imbi12d 311 . . . . . . . . 9 1c Nc Spac Tc Nc Spac Nn 1c Nc Spac Tc Nc Spac Nn
2221cbvralv 2835 . . . . . . . 8 NC 1c Nc Spac Tc Nc Spac Nn NC 1c Nc Spac Tc Nc Spac Nn
2313, 22syl6bb 252 . . . . . . 7 1c NC Nc Spac Tc Nc Spac Nn NC 1c Nc Spac Tc Nc Spac Nn
24 eqeq1 2359 . . . . . . . . 9 Nc Spac Tc Nc Spac Tc
2524imbi1d 308 . . . . . . . 8 Nc Spac Tc Nc Spac Nn Nc Spac Tc Nc Spac Nn
2625ralbidv 2634 . . . . . . 7 NC Nc Spac Tc Nc Spac Nn NC Nc Spac Tc Nc Spac Nn
27 tccl 6160 . . . . . . . . . . . . 13 NC Tc NC
28 te0c 6237 . . . . . . . . . . . . 13 NC Tc c 0c NC
29 nchoicelem7 6295 . . . . . . . . . . . . 13 Tc NC Tc c 0c NC Nc Spac Tc Nc Spac 2cc Tc 1c
3027, 28, 29syl2anc 642 . . . . . . . . . . . 12 NC Nc Spac Tc Nc Spac 2cc Tc 1c
31 0cnsuc 4401 . . . . . . . . . . . . 13 Nc Spac 2cc Tc 1c 0c
3231a1i 10 . . . . . . . . . . . 12 NC Nc Spac 2cc Tc 1c 0c
3330, 32eqnetrd 2534 . . . . . . . . . . 11 NC Nc Spac Tc 0c
3433necomd 2599 . . . . . . . . . 10 NC 0c Nc Spac Tc
35 df-ne 2518 . . . . . . . . . 10 0c Nc Spac Tc 0c Nc Spac Tc
3634, 35sylib 188 . . . . . . . . 9 NC 0c Nc Spac Tc
3736pm2.21d 98 . . . . . . . 8 NC 0c Nc Spac Tc Nc Spac Nn
3837rgen 2679 . . . . . . 7 NC 0c Nc Spac Tc Nc Spac Nn
39 2nnc 6167 . . . . . . . . . . . . . . . . 17 2c Nn
40 ceclnn1 6189 . . . . . . . . . . . . . . . . 17 2c Nn NC c 0c NC 2cc NC
4139, 40mp3an1 1264 . . . . . . . . . . . . . . . 16 NC c 0c NC 2cc NC
42 tceq 6158 . . . . . . . . . . . . . . . . . . . . 21 2cc Tc Tc 2cc
4342fveq2d 5332 . . . . . . . . . . . . . . . . . . . 20 2cc Spac Tc Spac Tc 2cc
4443nceqd 6110 . . . . . . . . . . . . . . . . . . 19 2cc Nc Spac Tc Nc Spac Tc 2cc
4544eqeq2d 2364 . . . . . . . . . . . . . . . . . 18 2cc Nc Spac Tc Nc Spac Tc 2cc
46 fveq2 5328 . . . . . . . . . . . . . . . . . . . 20 2cc Spac Spac 2cc
4746nceqd 6110 . . . . . . . . . . . . . . . . . . 19 2cc Nc Spac Nc Spac 2cc
4847eleq1d 2419 . . . . . . . . . . . . . . . . . 18 2cc Nc Spac Nn Nc Spac 2cc Nn
4945, 48imbi12d 311 . . . . . . . . . . . . . . . . 17 2cc Nc Spac Tc Nc Spac Nn Nc Spac Tc 2cc Nc Spac 2cc Nn
5049rspcv 2951 . . . . . . . . . . . . . . . 16 2cc NC NC Nc Spac Tc Nc Spac Nn Nc Spac Tc 2cc Nc Spac 2cc Nn
5141, 50syl 15 . . . . . . . . . . . . . . 15 NC c 0c NC NC Nc Spac Tc Nc Spac Nn Nc Spac Tc 2cc Nc Spac 2cc Nn
5251ancoms 439 . . . . . . . . . . . . . 14 c 0c NC NC NC Nc Spac Tc Nc Spac Nn Nc Spac Tc 2cc Nc Spac 2cc Nn
5352adantrl 696 . . . . . . . . . . . . 13 c 0c NC Nn NC NC Nc Spac Tc Nc Spac Nn Nc Spac Tc 2cc Nc Spac 2cc Nn
54 tccl 6160 . . . . . . . . . . . . . . . . . . . . . . 23 NC Tc NC
55 te0c 6237 . . . . . . . . . . . . . . . . . . . . . . 23 NC Tc c 0c NC
56 nchoicelem7 6295 . . . . . . . . . . . . . . . . . . . . . . 23 Tc NC Tc c 0c NC Nc Spac Tc Nc Spac 2cc Tc 1c
5754, 55, 56syl2anc 642 . . . . . . . . . . . . . . . . . . . . . 22 NC Nc Spac Tc Nc Spac 2cc Tc 1c
5857adantl 452 . . . . . . . . . . . . . . . . . . . . 21 Nn NC Nc Spac Tc Nc Spac 2cc Tc 1c
5958adantl 452 . . . . . . . . . . . . . . . . . . . 20 c 0c NC Nn NC Nc Spac Tc Nc Spac 2cc Tc 1c
6059eqeq2d 2364 . . . . . . . . . . . . . . . . . . 19 c 0c NC Nn NC 1c Nc Spac Tc 1c Nc Spac 2cc Tc 1c
61 nnnc 6146 . . . . . . . . . . . . . . . . . . . . . . 23 Nn NC
6261adantr 451 . . . . . . . . . . . . . . . . . . . . . 22 Nn NC NC
6362adantl 452 . . . . . . . . . . . . . . . . . . . . 21 c 0c NC Nn NC NC
64 fvex 5339 . . . . . . . . . . . . . . . . . . . . . 22 Spac 2cc Tc
6564ncelncsi 6121 . . . . . . . . . . . . . . . . . . . . 21 Nc Spac 2cc Tc NC
66 peano4nc 6150 . . . . . . . . . . . . . . . . . . . . 21 NC Nc Spac 2cc Tc NC 1c Nc Spac 2cc Tc 1c Nc Spac 2cc Tc
6763, 65, 66sylancl 643 . . . . . . . . . . . . . . . . . . . 20 c 0c NC Nn NC 1c Nc Spac 2cc Tc 1c Nc Spac 2cc Tc
68 tce2 6236 . . . . . . . . . . . . . . . . . . . . . . . . . 26 NC c 0c NC Tc 2cc 2cc Tc
6968ancoms 439 . . . . . . . . . . . . . . . . . . . . . . . . 25 c 0c NC NC Tc 2cc 2cc Tc
7069adantrl 696 . . . . . . . . . . . . . . . . . . . . . . . 24 c 0c NC Nn NC Tc 2cc 2cc Tc
7170fveq2d 5332 . . . . . . . . . . . . . . . . . . . . . . 23 c 0c NC Nn NC Spac Tc 2cc Spac 2cc Tc
7271nceqd 6110 . . . . . . . . . . . . . . . . . . . . . 22 c 0c NC Nn NC Nc Spac Tc 2cc Nc Spac 2cc Tc
7372eqeq2d 2364 . . . . . . . . . . . . . . . . . . . . 21 c 0c NC Nn NC Nc Spac Tc 2cc Nc Spac 2cc Tc
7473biimprd 214 . . . . . . . . . . . . . . . . . . . 20 c 0c NC Nn NC Nc Spac 2cc Tc Nc Spac Tc 2cc
7567, 74sylbid 206 . . . . . . . . . . . . . . . . . . 19 c 0c NC Nn NC 1c Nc Spac 2cc Tc 1c Nc Spac Tc 2cc
7660, 75sylbid 206 . . . . . . . . . . . . . . . . . 18 c 0c NC Nn NC 1c Nc Spac Tc Nc Spac Tc 2cc
7776imim1d 69 . . . . . . . . . . . . . . . . 17 c 0c NC Nn NC Nc Spac Tc 2cc Nc Spac 2cc Nn 1c Nc Spac Tc Nc Spac 2cc Nn
7877imp 418 . . . . . . . . . . . . . . . 16 c 0c NC Nn NC Nc Spac Tc 2cc Nc Spac 2cc Nn 1c Nc Spac Tc Nc Spac 2cc Nn
79 peano2 4403 . . . . . . . . . . . . . . . 16 Nc Spac 2cc Nn Nc Spac 2cc 1c Nn
8078, 79syl6 29 . . . . . . . . . . . . . . 15 c 0c NC Nn NC Nc Spac Tc 2cc Nc Spac 2cc Nn 1c Nc Spac Tc Nc Spac 2cc 1c Nn
81 nchoicelem7 6295 . . . . . . . . . . . . . . . . . . 19 NC c 0c NC Nc Spac Nc Spac 2cc 1c
8281ancoms 439 . . . . . . . . . . . . . . . . . 18 c 0c NC NC Nc Spac Nc Spac 2cc 1c
8382adantrl 696 . . . . . . . . . . . . . . . . 17 c 0c NC Nn NC Nc Spac Nc Spac 2cc 1c
8483adantr 451 . . . . . . . . . . . . . . . 16 c 0c NC Nn NC Nc Spac Tc 2cc Nc Spac 2cc Nn Nc Spac Nc Spac 2cc 1c
8584eleq1d 2419 . . . . . . . . . . . . . . 15 c 0c NC Nn NC Nc Spac Tc 2cc Nc Spac 2cc Nn Nc Spac Nn Nc Spac 2cc 1c Nn
8680, 85sylibrd 225 . . . . . . . . . . . . . 14 c 0c NC Nn NC Nc Spac Tc 2cc Nc Spac 2cc Nn 1c Nc Spac Tc Nc Spac Nn
8786ex 423 . . . . . . . . . . . . 13 c 0c NC Nn NC Nc Spac Tc 2cc Nc Spac 2cc Nn 1c Nc Spac Tc Nc Spac Nn
8853, 87syld 40 . . . . . . . . . . . 12 c 0c NC Nn NC NC Nc Spac Tc Nc Spac Nn 1c Nc Spac Tc Nc Spac Nn
8988expimpd 586 . . . . . . . . . . 11 c 0c NC Nn NC NC Nc Spac Tc Nc Spac Nn 1c Nc Spac Tc Nc Spac Nn
90 nchoicelem3 6291 . . . . . . . . . . . . . . . . 17 NC c 0c NC Spac
9190nceqd 6110 . . . . . . . . . . . . . . . 16 NC c 0c NC Nc Spac Nc
92 vex 2862 . . . . . . . . . . . . . . . . . 18
9392df1c3 6140 . . . . . . . . . . . . . . . . 17 1c Nc
94 1cnnc 4408 . . . . . . . . . . . . . . . . 17 1c Nn
9593, 94eqeltrri 2424 . . . . . . . . . . . . . . . 16 Nc Nn
9691, 95syl6eqel 2441 . . . . . . . . . . . . . . 15 NC c 0c NC Nc Spac Nn
9796a1d 22 . . . . . . . . . . . . . 14 NC c 0c NC 1c Nc Spac Tc Nc Spac Nn
9897expcom 424 . . . . . . . . . . . . 13 c 0c NC NC 1c Nc Spac Tc Nc Spac Nn
9998adantld 453 . . . . . . . . . . . 12 c 0c NC Nn NC 1c Nc Spac Tc Nc Spac Nn
10099adantrd 454 . . . . . . . . . . 11 c 0c NC Nn NC NC Nc Spac Tc Nc Spac Nn 1c Nc Spac Tc Nc Spac Nn
10189, 100pm2.61i 156 . . . . . . . . . 10 Nn NC NC Nc Spac Tc Nc Spac Nn 1c Nc Spac Tc Nc Spac Nn
102101an32s 779 . . . . . . . . 9 Nn NC Nc Spac Tc Nc Spac Nn NC 1c Nc Spac Tc Nc Spac Nn
103102ralrimiva 2697 . . . . . . . 8 Nn NC Nc Spac Tc Nc Spac Nn NC 1c Nc Spac Tc Nc Spac Nn
104103ex 423 . . . . . . 7 Nn NC Nc Spac Tc Nc Spac Nn NC 1c Nc Spac Tc Nc Spac Nn
1054, 7, 10, 23, 26, 38, 104finds 4411 . . . . . 6 Nn NC Nc Spac Tc Nc Spac Nn
106 tceq 6158 . . . . . . . . . . 11 Tc Tc
107106fveq2d 5332 . . . . . . . . . 10 Spac Tc Spac Tc
108107nceqd 6110 . . . . . . . . 9 Nc Spac Tc Nc Spac Tc
109108eqeq2d 2364 . . . . . . . 8 Nc Spac Tc Nc Spac Tc
110 fveq2 5328 . . . . . . . . . . 11 Spac Spac
111110nceqd 6110 . . . . . . . . . 10 Nc Spac Nc Spac
112111eleq1d 2419 . . . . . . . . 9 Nc Spac Nn Nc Spac Nn
113 finnc 6243 . . . . . . . . 9 Spac Fin Nc Spac Nn
114112, 113syl6bbr 254 . . . . . . . 8 Nc Spac Nn Spac Fin
115109, 114imbi12d 311 . . . . . . 7 Nc Spac Tc Nc Spac Nn Nc Spac Tc Spac Fin
116115rspccv 2952 . . . . . 6 NC Nc Spac Tc Nc Spac Nn NC Nc Spac Tc Spac Fin
117105, 116syl 15 . . . . 5 Nn NC Nc Spac Tc Spac Fin
118117com23 72 . . . 4 Nn Nc Spac Tc NC Spac Fin
119118rexlimiv 2732 . . 3 Nn Nc Spac Tc NC Spac Fin
1203, 119sylbi 187 . 2 Spac Tc Fin NC Spac Fin
121120impcom 419 1 NC Spac Tc Fin Spac Fin
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358   wceq 1642   wcel 1710   wne 2516  wral 2614  wrex 2615  csn 3737  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   cplc 4375   Fin cfin 4376  cfv 4781  (class class class)co 5525   NC cncs 6088   Nc cnc 6091   Tc ctc 6093  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-lec 6099  df-ltc 6100  df-nc 6101  df-tc 6103  df-2c 6104  df-ce 6106  df-tcfn 6107  df-spac 6274 This theorem is referenced by:  nchoicelem19  6307
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