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Theorem nchoicelem14 6302
 Description: Lemma for nchoice 6308. When the special set generator yields a singleton, then the cardinal is not raisable. (Contributed by SF, 19-Mar-2015.)
Assertion
Ref Expression
nchoicelem14 NC Nc Spac 1c c 0c NC

Proof of Theorem nchoicelem14
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nchoicelem5 6293 . . . . . . . . 9 NC c 0c NC Spac 2cc
2 incom 3448 . . . . . . . . . . 11 Spac 2cc Spac 2cc
32eqeq1i 2360 . . . . . . . . . 10 Spac 2cc Spac 2cc
4 disjsn 3786 . . . . . . . . . 10 Spac 2cc Spac 2cc
53, 4bitri 240 . . . . . . . . 9 Spac 2cc Spac 2cc
61, 5sylibr 203 . . . . . . . 8 NC c 0c NC Spac 2cc
7 snex 4111 . . . . . . . . 9
8 fvex 5339 . . . . . . . . 9 Spac 2cc
97, 8ncdisjun 6136 . . . . . . . 8 Spac 2cc Nc Spac 2cc Nc Nc Spac 2cc
106, 9syl 15 . . . . . . 7 NC c 0c NC Nc Spac 2cc Nc Nc Spac 2cc
11 df1c3g 6141 . . . . . . . . . . 11 NC 1c Nc
1211adantr 451 . . . . . . . . . 10 NC c 0c NC 1c Nc
1312addceq2d 4390 . . . . . . . . 9 NC c 0c NC Nc Spac 2cc 1c Nc Spac 2cc Nc
14 addccom 4406 . . . . . . . . 9 Nc Nc Spac 2cc Nc Spac 2cc Nc
1513, 14syl6reqr 2404 . . . . . . . 8 NC c 0c NC Nc Nc Spac 2cc Nc Spac 2cc 1c
16 2nnc 6167 . . . . . . . . . . 11 2c Nn
17 ceclnn1 6189 . . . . . . . . . . 11 2c Nn NC c 0c NC 2cc NC
1816, 17mp3an1 1264 . . . . . . . . . 10 NC c 0c NC 2cc NC
19 nchoicelem13 6301 . . . . . . . . . 10 2cc NC 1c c Nc Spac 2cc
20 1cnc 6139 . . . . . . . . . . . 12 1c NC
218ncelncsi 6121 . . . . . . . . . . . 12 Nc Spac 2cc NC
22 dflec2 6210 . . . . . . . . . . . 12 1c NC Nc Spac 2cc NC 1c c Nc Spac 2cc NC Nc Spac 2cc 1c
2320, 21, 22mp2an 653 . . . . . . . . . . 11 1c c Nc Spac 2cc NC Nc Spac 2cc 1c
24 addccom 4406 . . . . . . . . . . . . . 14 1c 1c
25 0cnsuc 4401 . . . . . . . . . . . . . 14 1c 0c
2624, 25eqnetri 2533 . . . . . . . . . . . . 13 1c 0c
27 neeq1 2524 . . . . . . . . . . . . 13 Nc Spac 2cc 1c Nc Spac 2cc 0c 1c 0c
2826, 27mpbiri 224 . . . . . . . . . . . 12 Nc Spac 2cc 1c Nc Spac 2cc 0c
2928rexlimivw 2734 . . . . . . . . . . 11 NC Nc Spac 2cc 1c Nc Spac 2cc 0c
3023, 29sylbi 187 . . . . . . . . . 10 1c c Nc Spac 2cc Nc Spac 2cc 0c
3118, 19, 303syl 18 . . . . . . . . 9 NC c 0c NC Nc Spac 2cc 0c
32 0cnc 6138 . . . . . . . . . . 11 0c NC
33 peano4nc 6150 . . . . . . . . . . 11 Nc Spac 2cc NC 0c NC Nc Spac 2cc 1c 0c 1c Nc Spac 2cc 0c
3421, 32, 33mp2an 653 . . . . . . . . . 10 Nc Spac 2cc 1c 0c 1c Nc Spac 2cc 0c
3534necon3bii 2548 . . . . . . . . 9 Nc Spac 2cc 1c 0c 1c Nc Spac 2cc 0c
3631, 35sylibr 203 . . . . . . . 8 NC c 0c NC Nc Spac 2cc 1c 0c 1c
3715, 36eqnetrd 2534 . . . . . . 7 NC c 0c NC Nc Nc Spac 2cc 0c 1c
3810, 37eqnetrd 2534 . . . . . 6 NC c 0c NC Nc Spac 2cc 0c 1c
39 addcid2 4407 . . . . . . . 8 0c 1c 1c
4039neeq2i 2527 . . . . . . 7 Nc Spac 2cc 0c 1c Nc Spac 2cc 1c
41 df-ne 2518 . . . . . . 7 Nc Spac 2cc 1c Nc Spac 2cc 1c
4240, 41bitri 240 . . . . . 6 Nc Spac 2cc 0c 1c Nc Spac 2cc 1c
4338, 42sylib 188 . . . . 5 NC c 0c NC Nc Spac 2cc 1c
44 nchoicelem6 6294 . . . . . . 7 NC c 0c NC Spac Spac 2cc
4544nceqd 6110 . . . . . 6 NC c 0c NC Nc Spac Nc Spac 2cc
4645eqeq1d 2361 . . . . 5 NC c 0c NC Nc Spac 1c Nc Spac 2cc 1c
4743, 46mtbird 292 . . . 4 NC c 0c NC Nc Spac 1c
4847ex 423 . . 3 NC c 0c NC Nc Spac 1c
4948con2d 107 . 2 NC Nc Spac 1c c 0c NC
5049imp 418 1 NC Nc Spac 1c c 0c NC
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358   wceq 1642   wcel 1710   wne 2516  wrex 2615   cun 3207   cin 3208  c0 3550  csn 3737  1cc1c 4134   Nn cnnc 4373  0cc0c 4374   cplc 4375   class class class wbr 4639  cfv 4781  (class class class)co 5525   NC cncs 6088   c clec 6089   Nc cnc 6091  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-lec 6099  df-ltc 6100  df-nc 6101  df-2c 6104  df-ce 6106  df-spac 6274 This theorem is referenced by:  nchoicelem17  6305
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