Theorem nchoicelem6 6295

Description: Lemma for nchoice 6309. 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 ∧ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) = ({M} ∪ ( Spac ‘(2c ↑c M))))

Proof of Theorem nchoicelem6

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 443	. . 3 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → M ∈ NC )
2		snex 4112	. . . . 5 ⊢ {M} ∈ V
3		fvex 5340	. . . . 5 ⊢ ( Spac ‘(2c ↑c M)) ∈ V
4	2, 3	unex 4107	. . . 4 ⊢ ({M} ∪ ( Spac ‘(2c ↑c M))) ∈ V
5	4	a1i 10	. . 3 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ({M} ∪ ( Spac ‘(2c ↑c M))) ∈ V)
6		snidg 3759	. . . . 5 ⊢ (M ∈ NC → M ∈ {M})
7	6	adantr 451	. . . 4 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → M ∈ {M})
8		elun1 3431	. . . 4 ⊢ (M ∈ {M} → M ∈ ({M} ∪ ( Spac ‘(2c ↑c M))))
9	7, 8	syl 15	. . 3 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → M ∈ ({M} ∪ ( Spac ‘(2c ↑c M))))
10		elun 3221	. . . . . . . . . 10 ⊢ (x ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) ↔ (x ∈ {M} ∨ x ∈ ( Spac ‘(2c ↑c M))))
11		elsn 3749	. . . . . . . . . . 11 ⊢ (x ∈ {M} ↔ x = M)
12	11	orbi1i 506	. . . . . . . . . 10 ⊢ ((x ∈ {M} ∨ x ∈ ( Spac ‘(2c ↑c M))) ↔ (x = M ∨ x ∈ ( Spac ‘(2c ↑c M))))
13	10, 12	bitri 240	. . . . . . . . 9 ⊢ (x ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) ↔ (x = M ∨ x ∈ ( Spac ‘(2c ↑c M))))
14		spacssnc 6285	. . . . . . . . . . . . . . 15 ⊢ (M ∈ NC → ( Spac ‘M) ⊆ NC )
15	14	adantr 451	. . . . . . . . . . . . . 14 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) ⊆ NC )
16		spacid 6286	. . . . . . . . . . . . . . . 16 ⊢ (M ∈ NC → M ∈ ( Spac ‘M))
17	16	adantr 451	. . . . . . . . . . . . . . 15 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → M ∈ ( Spac ‘M))
18		simpr 447	. . . . . . . . . . . . . . 15 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (M ↑c 0c) ∈ NC )
19		spaccl 6287	. . . . . . . . . . . . . . 15 ⊢ ((M ∈ NC ∧ M ∈ ( Spac ‘M) ∧ (M ↑c 0c) ∈ NC ) → (2c ↑c M) ∈ ( Spac ‘M))
20	1, 17, 18, 19	syl3anc 1182	. . . . . . . . . . . . . 14 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (2c ↑c M) ∈ ( Spac ‘M))
21	15, 20	sseldd 3275	. . . . . . . . . . . . 13 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (2c ↑c M) ∈ NC )
22		spacid 6286	. . . . . . . . . . . . 13 ⊢ ((2c ↑c M) ∈ NC → (2c ↑c M) ∈ ( Spac ‘(2c ↑c M)))
23	21, 22	syl 15	. . . . . . . . . . . 12 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (2c ↑c M) ∈ ( Spac ‘(2c ↑c M)))
24		oveq2 5532	. . . . . . . . . . . . 13 ⊢ (x = M → (2c ↑c x) = (2c ↑c M))
25	24	eleq1d 2419	. . . . . . . . . . . 12 ⊢ (x = M → ((2c ↑c x) ∈ ( Spac ‘(2c ↑c M)) ↔ (2c ↑c M) ∈ ( Spac ‘(2c ↑c M))))
26	23, 25	syl5ibrcom 213	. . . . . . . . . . 11 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (x = M → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M))))
27	26	adantr 451	. . . . . . . . . 10 ⊢ (((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) ∧ (x ↑c 0c) ∈ NC ) → (x = M → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M))))
28		2nnc 6168	. . . . . . . . . . . . . 14 ⊢ 2c ∈ Nn
29		ceclnn1 6190	. . . . . . . . . . . . . 14 ⊢ ((2c ∈ Nn ∧ M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (2c ↑c M) ∈ NC )
30	28, 29	mp3an1 1264	. . . . . . . . . . . . 13 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (2c ↑c M) ∈ NC )
31	30	adantr 451	. . . . . . . . . . . 12 ⊢ (((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) ∧ ((x ↑c 0c) ∈ NC ∧ x ∈ ( Spac ‘(2c ↑c M)))) → (2c ↑c M) ∈ NC )
32		simprr 733	. . . . . . . . . . . 12 ⊢ (((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) ∧ ((x ↑c 0c) ∈ NC ∧ x ∈ ( Spac ‘(2c ↑c M)))) → x ∈ ( Spac ‘(2c ↑c M)))
33		simprl 732	. . . . . . . . . . . 12 ⊢ (((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) ∧ ((x ↑c 0c) ∈ NC ∧ x ∈ ( Spac ‘(2c ↑c M)))) → (x ↑c 0c) ∈ NC )
34		spaccl 6287	. . . . . . . . . . . 12 ⊢ (((2c ↑c M) ∈ NC ∧ x ∈ ( Spac ‘(2c ↑c M)) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M)))
35	31, 32, 33, 34	syl3anc 1182	. . . . . . . . . . 11 ⊢ (((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) ∧ ((x ↑c 0c) ∈ NC ∧ x ∈ ( Spac ‘(2c ↑c M)))) → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M)))
36	35	expr 598	. . . . . . . . . 10 ⊢ (((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) ∧ (x ↑c 0c) ∈ NC ) → (x ∈ ( Spac ‘(2c ↑c M)) → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M))))
37	27, 36	jaod 369	. . . . . . . . 9 ⊢ (((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) ∧ (x ↑c 0c) ∈ NC ) → ((x = M ∨ x ∈ ( Spac ‘(2c ↑c M))) → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M))))
38	13, 37	syl5bi 208	. . . . . . . 8 ⊢ (((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) ∧ (x ↑c 0c) ∈ NC ) → (x ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M))))
39	38	ex 423	. . . . . . 7 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ((x ↑c 0c) ∈ NC → (x ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M)))))
40	39	com23 72	. . . . . 6 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (x ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) → ((x ↑c 0c) ∈ NC → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M)))))
41	40	imp3a 420	. . . . 5 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ((x ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ( Spac ‘(2c ↑c M))))
42		elun2 3432	. . . . 5 ⊢ ((2c ↑c x) ∈ ( Spac ‘(2c ↑c M)) → (2c ↑c x) ∈ ({M} ∪ ( Spac ‘(2c ↑c M))))
43	41, 42	syl6 29	. . . 4 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ((x ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ({M} ∪ ( Spac ‘(2c ↑c M)))))
44	43	ralrimivw 2699	. . 3 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ∀x ∈ ( Spac ‘M)((x ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ({M} ∪ ( Spac ‘(2c ↑c M)))))
45		spacind 6288	. . 3 ⊢ (((M ∈ NC ∧ ({M} ∪ ( Spac ‘(2c ↑c M))) ∈ V) ∧ (M ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) ∧ ∀x ∈ ( Spac ‘M)((x ∈ ({M} ∪ ( Spac ‘(2c ↑c M))) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ({M} ∪ ( Spac ‘(2c ↑c M)))))) → ( Spac ‘M) ⊆ ({M} ∪ ( Spac ‘(2c ↑c M))))
46	1, 5, 9, 44, 45	syl22anc 1183	. 2 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) ⊆ ({M} ∪ ( Spac ‘(2c ↑c M))))
47	16	snssd 3854	. . . 4 ⊢ (M ∈ NC → {M} ⊆ ( Spac ‘M))
48	47	adantr 451	. . 3 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → {M} ⊆ ( Spac ‘M))
49		fvex 5340	. . . . 5 ⊢ ( Spac ‘M) ∈ V
50	49	a1i 10	. . . 4 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) ∈ V)
51		spaccl 6287	. . . . . . 7 ⊢ ((M ∈ NC ∧ x ∈ ( Spac ‘M) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ( Spac ‘M))
52	51	3expib 1154	. . . . . 6 ⊢ (M ∈ NC → ((x ∈ ( Spac ‘M) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ( Spac ‘M)))
53	52	adantr 451	. . . . 5 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ((x ∈ ( Spac ‘M) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ( Spac ‘M)))
54	53	ralrimivw 2699	. . . 4 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ∀x ∈ ( Spac ‘(2c ↑c M))((x ∈ ( Spac ‘M) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ( Spac ‘M)))
55		spacind 6288	. . . 4 ⊢ ((((2c ↑c M) ∈ NC ∧ ( Spac ‘M) ∈ V) ∧ ((2c ↑c M) ∈ ( Spac ‘M) ∧ ∀x ∈ ( Spac ‘(2c ↑c M))((x ∈ ( Spac ‘M) ∧ (x ↑c 0c) ∈ NC ) → (2c ↑c x) ∈ ( Spac ‘M)))) → ( Spac ‘(2c ↑c M)) ⊆ ( Spac ‘M))
56	21, 50, 20, 54, 55	syl22anc 1183	. . 3 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Spac ‘(2c ↑c M)) ⊆ ( Spac ‘M))
57	48, 56	unssd 3440	. 2 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ({M} ∪ ( Spac ‘(2c ↑c M))) ⊆ ( Spac ‘M))
58	46, 57	eqssd 3290	1 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) = ({M} ∪ ( Spac ‘(2c ↑c M))))

This proof depends on syntax axioms:   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860   ∪ cun 3208   ⊆ wss 3258  {csn 3738   Nn cnnc 4374  0_cc0c 4375   ‘cfv 4782  (class class class)co 5526   NC cncs 6089  2_cc2c 6095   ↑_c cce 6097   Sp_ac cspac 6274

This proof depends on 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-fullfun 5769  df-clos1 5874  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-2c 6105  df-ce 6107  df-spac 6275

This theorem is used by:  nchoicelem7  6296  nchoicelem14  6303