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Theorem nchoicelem8 6296
 Description: Lemma for nchoice 6308. An anti-closure condition for cardinal exponentiation to zero. Theorem 4.5 of [Specker] p. 973. (Contributed by SF, 18-Mar-2015.)
Assertion
Ref Expression
nchoicelem8 c We NC NC c 0c NC Nc 1c c

Proof of Theorem nchoicelem8
StepHypRef Expression
1 ce0lenc1 6239 . . . 4 NC c 0c NC c Nc 1c
21notbid 285 . . 3 NC c 0c NC c Nc 1c
32adantl 452 . 2 c We NC NC c 0c NC c Nc 1c
4 df-we 5906 . . . . . . . . . 10 We Or Fr
54breqi 4645 . . . . . . . . 9 c We NC c Or Fr NC
6 brin 4693 . . . . . . . . 9 c Or Fr NC c Or NC c Fr NC
75, 6bitri 240 . . . . . . . 8 c We NC c Or NC c Fr NC
8 sopc 5934 . . . . . . . . . 10 c Or NC c Po NC c Connex NC
98simprbi 450 . . . . . . . . 9 c Or NC c Connex NC
109adantr 451 . . . . . . . 8 c Or NC c Fr NC c Connex NC
117, 10sylbi 187 . . . . . . 7 c We NC c Connex NC
12 simpl 443 . . . . . . . 8 c Connex NC NC c Connex NC
13 simpr 447 . . . . . . . 8 c Connex NC NC NC
14 1cex 4142 . . . . . . . . . 10 1c
1514ncelncsi 6121 . . . . . . . . 9 Nc 1c NC
1615a1i 10 . . . . . . . 8 c Connex NC NC Nc 1c NC
1712, 13, 16connexd 5931 . . . . . . 7 c Connex NC NC c Nc 1c Nc 1c c
1811, 17sylan 457 . . . . . 6 c We NC NC c Nc 1c Nc 1c c
1918ord 366 . . . . 5 c We NC NC c Nc 1c Nc 1c c
20 id 19 . . . . . . . 8 Nc 1c Nc 1c
21 nclecid 6197 . . . . . . . . 9 Nc 1c NC Nc 1c c Nc 1c
2215, 21ax-mp 8 . . . . . . . 8 Nc 1c c Nc 1c
2320, 22syl6eqbrr 4677 . . . . . . 7 Nc 1c c Nc 1c
2423necon3bi 2557 . . . . . 6 c Nc 1c Nc 1c
2524a1i 10 . . . . 5 c We NC NC c Nc 1c Nc 1c
2619, 25jcad 519 . . . 4 c We NC NC c Nc 1c Nc 1c c Nc 1c
277simplbi 446 . . . . . . . . . . 11 c We NC c Or NC
288simplbi 446 . . . . . . . . . . 11 c Or NC c Po NC
29 df-partial 5902 . . . . . . . . . . . . . 14 Po Ref Trans Antisym
3029breqi 4645 . . . . . . . . . . . . 13 c Po NC c Ref Trans Antisym NC
31 brin 4693 . . . . . . . . . . . . 13 c Ref Trans Antisym NC c Ref Trans NC c Antisym NC
3230, 31bitri 240 . . . . . . . . . . . 12 c Po NC c Ref Trans NC c Antisym NC
3332simprbi 450 . . . . . . . . . . 11 c Po NC c Antisym NC
3427, 28, 333syl 18 . . . . . . . . . 10 c We NC c Antisym NC
3534adantr 451 . . . . . . . . 9 c We NC NC c Antisym NC
3635adantr 451 . . . . . . . 8 c We NC NC Nc 1c c c Nc 1c c Antisym NC
3715a1i 10 . . . . . . . 8 c We NC NC Nc 1c c c Nc 1c Nc 1c NC
38 simplr 731 . . . . . . . 8 c We NC NC Nc 1c c c Nc 1c NC
39 simprl 732 . . . . . . . 8 c We NC NC Nc 1c c c Nc 1c Nc 1c c
40 simprr 733 . . . . . . . 8 c We NC NC Nc 1c c c Nc 1c c Nc 1c
4136, 37, 38, 39, 40antid 5929 . . . . . . 7 c We NC NC Nc 1c c c Nc 1c Nc 1c
4241expr 598 . . . . . 6 c We NC NC Nc 1c c c Nc 1c Nc 1c
4342necon3ad 2552 . . . . 5 c We NC NC Nc 1c c Nc 1c c Nc 1c
4443expimpd 586 . . . 4 c We NC NC Nc 1c c Nc 1c c Nc 1c
4526, 44impbid 183 . . 3 c We NC NC c Nc 1c Nc 1c c Nc 1c
46 brltc 6114 . . 3 Nc 1c c Nc 1c c Nc 1c
4745, 46syl6bbr 254 . 2 c We NC NC c Nc 1c Nc 1c c
483, 47bitrd 244 1 c We NC NC c 0c NC Nc 1c c
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wo 357   wa 358   wceq 1642   wcel 1710   wne 2516   cin 3208  1cc1c 4134  0cc0c 4374   class class class wbr 4639  (class class class)co 5525   Trans ctrans 5888   Ref cref 5889   Antisym cantisym 5890   Po cpartial 5891   Connex cconnex 5892   Or cstrict 5893   Fr cfound 5894   We cwe 5895   NC cncs 6088   c clec 6089   c cltc 6090   Nc cnc 6091   ↑c cce 6096 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-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-trans 5899  df-antisym 5901  df-partial 5902  df-connex 5903  df-strict 5904  df-we 5906  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-ce 6106 This theorem is referenced by:  nchoicelem9  6297  nchoicelem19  6307
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