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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 ((M NC Nc ( SpacM) = 1c) → ¬ (Mc 0c) NC )

Proof of Theorem nchoicelem14
Dummy variable k is distinct from all other variables.
StepHypRef Expression
1 nchoicelem5 6293 . . . . . . . . 9 ((M NC (Mc 0c) NC ) → ¬ M ( Spac ‘(2cc M)))
2 incom 3448 . . . . . . . . . . 11 ({M} ∩ ( Spac ‘(2cc M))) = (( Spac ‘(2cc M)) ∩ {M})
32eqeq1i 2360 . . . . . . . . . 10 (({M} ∩ ( Spac ‘(2cc M))) = ↔ (( Spac ‘(2cc M)) ∩ {M}) = )
4 disjsn 3786 . . . . . . . . . 10 ((( Spac ‘(2cc M)) ∩ {M}) = ↔ ¬ M ( Spac ‘(2cc M)))
53, 4bitri 240 . . . . . . . . 9 (({M} ∩ ( Spac ‘(2cc M))) = ↔ ¬ M ( Spac ‘(2cc M)))
61, 5sylibr 203 . . . . . . . 8 ((M NC (Mc 0c) NC ) → ({M} ∩ ( Spac ‘(2cc M))) = )
7 snex 4111 . . . . . . . . 9 {M} V
8 fvex 5339 . . . . . . . . 9 ( Spac ‘(2cc M)) V
97, 8ncdisjun 6136 . . . . . . . 8 (({M} ∩ ( Spac ‘(2cc M))) = Nc ({M} ∪ ( Spac ‘(2cc M))) = ( Nc {M} +c Nc ( Spac ‘(2cc M))))
106, 9syl 15 . . . . . . 7 ((M NC (Mc 0c) NC ) → Nc ({M} ∪ ( Spac ‘(2cc M))) = ( Nc {M} +c Nc ( Spac ‘(2cc M))))
11 df1c3g 6141 . . . . . . . . . . 11 (M NC → 1c = Nc {M})
1211adantr 451 . . . . . . . . . 10 ((M NC (Mc 0c) NC ) → 1c = Nc {M})
1312addceq2d 4390 . . . . . . . . 9 ((M NC (Mc 0c) NC ) → ( Nc ( Spac ‘(2cc M)) +c 1c) = ( Nc ( Spac ‘(2cc M)) +c Nc {M}))
14 addccom 4406 . . . . . . . . 9 ( Nc {M} +c Nc ( Spac ‘(2cc M))) = ( Nc ( Spac ‘(2cc M)) +c Nc {M})
1513, 14syl6reqr 2404 . . . . . . . 8 ((M NC (Mc 0c) NC ) → ( Nc {M} +c Nc ( Spac ‘(2cc M))) = ( Nc ( Spac ‘(2cc M)) +c 1c))
16 2nnc 6167 . . . . . . . . . . 11 2c Nn
17 ceclnn1 6189 . . . . . . . . . . 11 ((2c Nn M NC (Mc 0c) NC ) → (2cc M) NC )
1816, 17mp3an1 1264 . . . . . . . . . 10 ((M NC (Mc 0c) NC ) → (2cc M) NC )
19 nchoicelem13 6301 . . . . . . . . . 10 ((2cc M) NC → 1cc Nc ( Spac ‘(2cc M)))
20 1cnc 6139 . . . . . . . . . . . 12 1c NC
218ncelncsi 6121 . . . . . . . . . . . 12 Nc ( Spac ‘(2cc M)) NC
22 dflec2 6210 . . . . . . . . . . . 12 ((1c NC Nc ( Spac ‘(2cc M)) NC ) → (1cc Nc ( Spac ‘(2cc M)) ↔ k NC Nc ( Spac ‘(2cc M)) = (1c +c k)))
2320, 21, 22mp2an 653 . . . . . . . . . . 11 (1cc Nc ( Spac ‘(2cc M)) ↔ k NC Nc ( Spac ‘(2cc M)) = (1c +c k))
24 addccom 4406 . . . . . . . . . . . . . 14 (1c +c k) = (k +c 1c)
25 0cnsuc 4401 . . . . . . . . . . . . . 14 (k +c 1c) ≠ 0c
2624, 25eqnetri 2533 . . . . . . . . . . . . 13 (1c +c k) ≠ 0c
27 neeq1 2524 . . . . . . . . . . . . 13 ( Nc ( Spac ‘(2cc M)) = (1c +c k) → ( Nc ( Spac ‘(2cc M)) ≠ 0c ↔ (1c +c k) ≠ 0c))
2826, 27mpbiri 224 . . . . . . . . . . . 12 ( Nc ( Spac ‘(2cc M)) = (1c +c k) → Nc ( Spac ‘(2cc M)) ≠ 0c)
2928rexlimivw 2734 . . . . . . . . . . 11 (k NC Nc ( Spac ‘(2cc M)) = (1c +c k) → Nc ( Spac ‘(2cc M)) ≠ 0c)
3023, 29sylbi 187 . . . . . . . . . 10 (1cc Nc ( Spac ‘(2cc M)) → Nc ( Spac ‘(2cc M)) ≠ 0c)
3118, 19, 303syl 18 . . . . . . . . 9 ((M NC (Mc 0c) NC ) → Nc ( Spac ‘(2cc M)) ≠ 0c)
32 0cnc 6138 . . . . . . . . . . 11 0c NC
33 peano4nc 6150 . . . . . . . . . . 11 (( Nc ( Spac ‘(2cc M)) NC 0c NC ) → (( Nc ( Spac ‘(2cc M)) +c 1c) = (0c +c 1c) ↔ Nc ( Spac ‘(2cc M)) = 0c))
3421, 32, 33mp2an 653 . . . . . . . . . 10 (( Nc ( Spac ‘(2cc M)) +c 1c) = (0c +c 1c) ↔ Nc ( Spac ‘(2cc M)) = 0c)
3534necon3bii 2548 . . . . . . . . 9 (( Nc ( Spac ‘(2cc M)) +c 1c) ≠ (0c +c 1c) ↔ Nc ( Spac ‘(2cc M)) ≠ 0c)
3631, 35sylibr 203 . . . . . . . 8 ((M NC (Mc 0c) NC ) → ( Nc ( Spac ‘(2cc M)) +c 1c) ≠ (0c +c 1c))
3715, 36eqnetrd 2534 . . . . . . 7 ((M NC (Mc 0c) NC ) → ( Nc {M} +c Nc ( Spac ‘(2cc M))) ≠ (0c +c 1c))
3810, 37eqnetrd 2534 . . . . . 6 ((M NC (Mc 0c) NC ) → Nc ({M} ∪ ( Spac ‘(2cc M))) ≠ (0c +c 1c))
39 addcid2 4407 . . . . . . . 8 (0c +c 1c) = 1c
4039neeq2i 2527 . . . . . . 7 ( Nc ({M} ∪ ( Spac ‘(2cc M))) ≠ (0c +c 1c) ↔ Nc ({M} ∪ ( Spac ‘(2cc M))) ≠ 1c)
41 df-ne 2518 . . . . . . 7 ( Nc ({M} ∪ ( Spac ‘(2cc M))) ≠ 1c ↔ ¬ Nc ({M} ∪ ( Spac ‘(2cc M))) = 1c)
4240, 41bitri 240 . . . . . 6 ( Nc ({M} ∪ ( Spac ‘(2cc M))) ≠ (0c +c 1c) ↔ ¬ Nc ({M} ∪ ( Spac ‘(2cc M))) = 1c)
4338, 42sylib 188 . . . . 5 ((M NC (Mc 0c) NC ) → ¬ Nc ({M} ∪ ( Spac ‘(2cc M))) = 1c)
44 nchoicelem6 6294 . . . . . . 7 ((M NC (Mc 0c) NC ) → ( SpacM) = ({M} ∪ ( Spac ‘(2cc M))))
4544nceqd 6110 . . . . . 6 ((M NC (Mc 0c) NC ) → Nc ( SpacM) = Nc ({M} ∪ ( Spac ‘(2cc M))))
4645eqeq1d 2361 . . . . 5 ((M NC (Mc 0c) NC ) → ( Nc ( SpacM) = 1cNc ({M} ∪ ( Spac ‘(2cc M))) = 1c))
4743, 46mtbird 292 . . . 4 ((M NC (Mc 0c) NC ) → ¬ Nc ( SpacM) = 1c)
4847ex 423 . . 3 (M NC → ((Mc 0c) NC → ¬ Nc ( SpacM) = 1c))
4948con2d 107 . 2 (M NC → ( Nc ( SpacM) = 1c → ¬ (Mc 0c) NC ))
5049imp 418 1 ((M NC Nc ( SpacM) = 1c) → ¬ (Mc 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   +c 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-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-2c 6104  df-ce 6106  df-spac 6274 This theorem is referenced by:  nchoicelem17  6305
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