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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 ((M NC (Mc 0c) NC ) → ( SpacM) = ({M} ∪ ( Spac ‘(2cc M))))

Proof of Theorem nchoicelem6
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3 ((M NC (Mc 0c) NC ) → M NC )
2 snex 4111 . . . . 5 {M} V
3 fvex 5339 . . . . 5 ( Spac ‘(2cc M)) V
42, 3unex 4106 . . . 4 ({M} ∪ ( Spac ‘(2cc M))) V
54a1i 10 . . 3 ((M NC (Mc 0c) NC ) → ({M} ∪ ( Spac ‘(2cc M))) V)
6 snidg 3758 . . . . 5 (M NCM {M})
76adantr 451 . . . 4 ((M NC (Mc 0c) NC ) → M {M})
8 elun1 3430 . . . 4 (M {M} → M ({M} ∪ ( Spac ‘(2cc M))))
97, 8syl 15 . . 3 ((M NC (Mc 0c) NC ) → M ({M} ∪ ( Spac ‘(2cc M))))
10 elun 3220 . . . . . . . . . 10 (x ({M} ∪ ( Spac ‘(2cc M))) ↔ (x {M} x ( Spac ‘(2cc M))))
11 elsn 3748 . . . . . . . . . . 11 (x {M} ↔ x = M)
1211orbi1i 506 . . . . . . . . . 10 ((x {M} x ( Spac ‘(2cc M))) ↔ (x = M x ( Spac ‘(2cc M))))
1310, 12bitri 240 . . . . . . . . 9 (x ({M} ∪ ( Spac ‘(2cc M))) ↔ (x = M x ( Spac ‘(2cc M))))
14 spacssnc 6284 . . . . . . . . . . . . . . 15 (M NC → ( SpacM) NC )
1514adantr 451 . . . . . . . . . . . . . 14 ((M NC (Mc 0c) NC ) → ( SpacM) NC )
16 spacid 6285 . . . . . . . . . . . . . . . 16 (M NCM ( SpacM))
1716adantr 451 . . . . . . . . . . . . . . 15 ((M NC (Mc 0c) NC ) → M ( SpacM))
18 simpr 447 . . . . . . . . . . . . . . 15 ((M NC (Mc 0c) NC ) → (Mc 0c) NC )
19 spaccl 6286 . . . . . . . . . . . . . . 15 ((M NC M ( SpacM) (Mc 0c) NC ) → (2cc M) ( SpacM))
201, 17, 18, 19syl3anc 1182 . . . . . . . . . . . . . 14 ((M NC (Mc 0c) NC ) → (2cc M) ( SpacM))
2115, 20sseldd 3274 . . . . . . . . . . . . 13 ((M NC (Mc 0c) NC ) → (2cc M) NC )
22 spacid 6285 . . . . . . . . . . . . 13 ((2cc M) NC → (2cc M) ( Spac ‘(2cc M)))
2321, 22syl 15 . . . . . . . . . . . 12 ((M NC (Mc 0c) NC ) → (2cc M) ( Spac ‘(2cc M)))
24 oveq2 5531 . . . . . . . . . . . . 13 (x = M → (2cc x) = (2cc M))
2524eleq1d 2419 . . . . . . . . . . . 12 (x = M → ((2cc x) ( Spac ‘(2cc M)) ↔ (2cc M) ( Spac ‘(2cc M))))
2623, 25syl5ibrcom 213 . . . . . . . . . . 11 ((M NC (Mc 0c) NC ) → (x = M → (2cc x) ( Spac ‘(2cc M))))
2726adantr 451 . . . . . . . . . 10 (((M NC (Mc 0c) NC ) (xc 0c) NC ) → (x = M → (2cc x) ( Spac ‘(2cc M))))
28 2nnc 6167 . . . . . . . . . . . . . 14 2c Nn
29 ceclnn1 6189 . . . . . . . . . . . . . 14 ((2c Nn M NC (Mc 0c) NC ) → (2cc M) NC )
3028, 29mp3an1 1264 . . . . . . . . . . . . 13 ((M NC (Mc 0c) NC ) → (2cc M) NC )
3130adantr 451 . . . . . . . . . . . 12 (((M NC (Mc 0c) NC ) ((xc 0c) NC x ( Spac ‘(2cc M)))) → (2cc M) NC )
32 simprr 733 . . . . . . . . . . . 12 (((M NC (Mc 0c) NC ) ((xc 0c) NC x ( Spac ‘(2cc M)))) → x ( Spac ‘(2cc M)))
33 simprl 732 . . . . . . . . . . . 12 (((M NC (Mc 0c) NC ) ((xc 0c) NC x ( Spac ‘(2cc M)))) → (xc 0c) NC )
34 spaccl 6286 . . . . . . . . . . . 12 (((2cc M) NC x ( Spac ‘(2cc M)) (xc 0c) NC ) → (2cc x) ( Spac ‘(2cc M)))
3531, 32, 33, 34syl3anc 1182 . . . . . . . . . . 11 (((M NC (Mc 0c) NC ) ((xc 0c) NC x ( Spac ‘(2cc M)))) → (2cc x) ( Spac ‘(2cc M)))
3635expr 598 . . . . . . . . . 10 (((M NC (Mc 0c) NC ) (xc 0c) NC ) → (x ( Spac ‘(2cc M)) → (2cc x) ( Spac ‘(2cc M))))
3727, 36jaod 369 . . . . . . . . 9 (((M NC (Mc 0c) NC ) (xc 0c) NC ) → ((x = M x ( Spac ‘(2cc M))) → (2cc x) ( Spac ‘(2cc M))))
3813, 37syl5bi 208 . . . . . . . 8 (((M NC (Mc 0c) NC ) (xc 0c) NC ) → (x ({M} ∪ ( Spac ‘(2cc M))) → (2cc x) ( Spac ‘(2cc M))))
3938ex 423 . . . . . . 7 ((M NC (Mc 0c) NC ) → ((xc 0c) NC → (x ({M} ∪ ( Spac ‘(2cc M))) → (2cc x) ( Spac ‘(2cc M)))))
4039com23 72 . . . . . 6 ((M NC (Mc 0c) NC ) → (x ({M} ∪ ( Spac ‘(2cc M))) → ((xc 0c) NC → (2cc x) ( Spac ‘(2cc M)))))
4140imp3a 420 . . . . 5 ((M NC (Mc 0c) NC ) → ((x ({M} ∪ ( Spac ‘(2cc M))) (xc 0c) NC ) → (2cc x) ( Spac ‘(2cc M))))
42 elun2 3431 . . . . 5 ((2cc x) ( Spac ‘(2cc M)) → (2cc x) ({M} ∪ ( Spac ‘(2cc M))))
4341, 42syl6 29 . . . 4 ((M NC (Mc 0c) NC ) → ((x ({M} ∪ ( Spac ‘(2cc M))) (xc 0c) NC ) → (2cc x) ({M} ∪ ( Spac ‘(2cc M)))))
4443ralrimivw 2698 . . 3 ((M NC (Mc 0c) NC ) → x ( SpacM)((x ({M} ∪ ( Spac ‘(2cc M))) (xc 0c) NC ) → (2cc x) ({M} ∪ ( Spac ‘(2cc M)))))
45 spacind 6287 . . 3 (((M NC ({M} ∪ ( Spac ‘(2cc M))) V) (M ({M} ∪ ( Spac ‘(2cc M))) x ( SpacM)((x ({M} ∪ ( Spac ‘(2cc M))) (xc 0c) NC ) → (2cc x) ({M} ∪ ( Spac ‘(2cc M)))))) → ( SpacM) ({M} ∪ ( Spac ‘(2cc M))))
461, 5, 9, 44, 45syl22anc 1183 . 2 ((M NC (Mc 0c) NC ) → ( SpacM) ({M} ∪ ( Spac ‘(2cc M))))
4716snssd 3853 . . . 4 (M NC → {M} ( SpacM))
4847adantr 451 . . 3 ((M NC (Mc 0c) NC ) → {M} ( SpacM))
49 fvex 5339 . . . . 5 ( SpacM) V
5049a1i 10 . . . 4 ((M NC (Mc 0c) NC ) → ( SpacM) V)
51 spaccl 6286 . . . . . . 7 ((M NC x ( SpacM) (xc 0c) NC ) → (2cc x) ( SpacM))
52513expib 1154 . . . . . 6 (M NC → ((x ( SpacM) (xc 0c) NC ) → (2cc x) ( SpacM)))
5352adantr 451 . . . . 5 ((M NC (Mc 0c) NC ) → ((x ( SpacM) (xc 0c) NC ) → (2cc x) ( SpacM)))
5453ralrimivw 2698 . . . 4 ((M NC (Mc 0c) NC ) → x ( Spac ‘(2cc M))((x ( SpacM) (xc 0c) NC ) → (2cc x) ( SpacM)))
55 spacind 6287 . . . 4 ((((2cc M) NC ( SpacM) V) ((2cc M) ( SpacM) x ( Spac ‘(2cc M))((x ( SpacM) (xc 0c) NC ) → (2cc x) ( SpacM)))) → ( Spac ‘(2cc M)) ( SpacM))
5621, 50, 20, 54, 55syl22anc 1183 . . 3 ((M NC (Mc 0c) NC ) → ( Spac ‘(2cc M)) ( SpacM))
5748, 56unssd 3439 . 2 ((M NC (Mc 0c) NC ) → ({M} ∪ ( Spac ‘(2cc M))) ( SpacM))
5846, 57eqssd 3289 1 ((M NC (Mc 0c) NC ) → ( SpacM) = ({M} ∪ ( Spac ‘(2cc M))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 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-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-nc 6101  df-2c 6104  df-ce 6106  df-spac 6274 This theorem is referenced by:  nchoicelem7  6295  nchoicelem14  6302
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