Theorem nchoicelem12 6301

Description: Lemma for nchoice 6309. If the T-raising of a cardinal yields a finite special set, then so does the initial set. Theorem 7.1 of [Specker] p. 974. (Contributed by SF, 18-Mar-2015.)

Assertion

Ref	Expression
nchoicelem12	⊢ ((M ∈ NC ∧ ( Spac ‘ Tc M) ∈ Fin ) → ( Spac ‘M) ∈ Fin )

Proof of Theorem nchoicelem12

Dummy variables m t x k n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		finnc 6244	. . . 4 ⊢ (( Spac ‘ Tc M) ∈ Fin ↔ Nc ( Spac ‘ Tc M) ∈ Nn )
2		risset 2662	. . . 4 ⊢ ( Nc ( Spac ‘ Tc M) ∈ Nn ↔ ∃x ∈ Nn x = Nc ( Spac ‘ Tc M))
3	1, 2	bitri 240	. . 3 ⊢ (( Spac ‘ Tc M) ∈ Fin ↔ ∃x ∈ Nn x = Nc ( Spac ‘ Tc M))
4		nchoicelem11 6300	. . . . . . 7 ⊢ {t ∣ ∀m ∈ NC (t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )} ∈ V
5		eqeq1 2359	. . . . . . . . 9 ⊢ (t = 0c → (t = Nc ( Spac ‘ Tc m) ↔ 0c = Nc ( Spac ‘ Tc m)))
6	5	imbi1d 308	. . . . . . . 8 ⊢ (t = 0c → ((t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ (0c = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )))
7	6	ralbidv 2635	. . . . . . 7 ⊢ (t = 0c → (∀m ∈ NC (t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ ∀m ∈ NC (0c = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )))
8		eqeq1 2359	. . . . . . . . 9 ⊢ (t = n → (t = Nc ( Spac ‘ Tc m) ↔ n = Nc ( Spac ‘ Tc m)))
9	8	imbi1d 308	. . . . . . . 8 ⊢ (t = n → ((t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )))
10	9	ralbidv 2635	. . . . . . 7 ⊢ (t = n → (∀m ∈ NC (t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ ∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )))
11		eqeq1 2359	. . . . . . . . . 10 ⊢ (t = (n +c 1c) → (t = Nc ( Spac ‘ Tc m) ↔ (n +c 1c) = Nc ( Spac ‘ Tc m)))
12	11	imbi1d 308	. . . . . . . . 9 ⊢ (t = (n +c 1c) → ((t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ ((n +c 1c) = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )))
13	12	ralbidv 2635	. . . . . . . 8 ⊢ (t = (n +c 1c) → (∀m ∈ NC (t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ ∀m ∈ NC ((n +c 1c) = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )))
14		tceq 6159	. . . . . . . . . . . . 13 ⊢ (m = k → Tc m = Tc k)
15	14	fveq2d 5333	. . . . . . . . . . . 12 ⊢ (m = k → ( Spac ‘ Tc m) = ( Spac ‘ Tc k))
16	15	nceqd 6111	. . . . . . . . . . 11 ⊢ (m = k → Nc ( Spac ‘ Tc m) = Nc ( Spac ‘ Tc k))
17	16	eqeq2d 2364	. . . . . . . . . 10 ⊢ (m = k → ((n +c 1c) = Nc ( Spac ‘ Tc m) ↔ (n +c 1c) = Nc ( Spac ‘ Tc k)))
18		fveq2 5329	. . . . . . . . . . . 12 ⊢ (m = k → ( Spac ‘m) = ( Spac ‘k))
19	18	nceqd 6111	. . . . . . . . . . 11 ⊢ (m = k → Nc ( Spac ‘m) = Nc ( Spac ‘k))
20	19	eleq1d 2419	. . . . . . . . . 10 ⊢ (m = k → ( Nc ( Spac ‘m) ∈ Nn ↔ Nc ( Spac ‘k) ∈ Nn ))
21	17, 20	imbi12d 311	. . . . . . . . 9 ⊢ (m = k → (((n +c 1c) = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn )))
22	21	cbvralv 2836	. . . . . . . 8 ⊢ (∀m ∈ NC ((n +c 1c) = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ ∀k ∈ NC ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn ))
23	13, 22	syl6bb 252	. . . . . . 7 ⊢ (t = (n +c 1c) → (∀m ∈ NC (t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ ∀k ∈ NC ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn )))
24		eqeq1 2359	. . . . . . . . 9 ⊢ (t = x → (t = Nc ( Spac ‘ Tc m) ↔ x = Nc ( Spac ‘ Tc m)))
25	24	imbi1d 308	. . . . . . . 8 ⊢ (t = x → ((t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ (x = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )))
26	25	ralbidv 2635	. . . . . . 7 ⊢ (t = x → (∀m ∈ NC (t = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ ∀m ∈ NC (x = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )))
27		tccl 6161	. . . . . . . . . . . . 13 ⊢ (m ∈ NC → Tc m ∈ NC )
28		te0c 6238	. . . . . . . . . . . . 13 ⊢ (m ∈ NC → ( Tc m ↑c 0c) ∈ NC )
29		nchoicelem7 6296	. . . . . . . . . . . . 13 ⊢ (( Tc m ∈ NC ∧ ( Tc m ↑c 0c) ∈ NC ) → Nc ( Spac ‘ Tc m) = ( Nc ( Spac ‘(2c ↑c Tc m)) +c 1c))
30	27, 28, 29	syl2anc 642	. . . . . . . . . . . 12 ⊢ (m ∈ NC → Nc ( Spac ‘ Tc m) = ( Nc ( Spac ‘(2c ↑c Tc m)) +c 1c))
31		0cnsuc 4402	. . . . . . . . . . . . 13 ⊢ ( Nc ( Spac ‘(2c ↑c Tc m)) +c 1c) ≠ 0c
32	31	a1i 10	. . . . . . . . . . . 12 ⊢ (m ∈ NC → ( Nc ( Spac ‘(2c ↑c Tc m)) +c 1c) ≠ 0c)
33	30, 32	eqnetrd 2535	. . . . . . . . . . 11 ⊢ (m ∈ NC → Nc ( Spac ‘ Tc m) ≠ 0c)
34	33	necomd 2600	. . . . . . . . . 10 ⊢ (m ∈ NC → 0c ≠ Nc ( Spac ‘ Tc m))
35		df-ne 2519	. . . . . . . . . 10 ⊢ (0c ≠ Nc ( Spac ‘ Tc m) ↔ ¬ 0c = Nc ( Spac ‘ Tc m))
36	34, 35	sylib 188	. . . . . . . . 9 ⊢ (m ∈ NC → ¬ 0c = Nc ( Spac ‘ Tc m))
37	36	pm2.21d 98	. . . . . . . 8 ⊢ (m ∈ NC → (0c = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ))
38	37	rgen 2680	. . . . . . 7 ⊢ ∀m ∈ NC (0c = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )
39		2nnc 6168	. . . . . . . . . . . . . . . . 17 ⊢ 2c ∈ Nn
40		ceclnn1 6190	. . . . . . . . . . . . . . . . 17 ⊢ ((2c ∈ Nn ∧ k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → (2c ↑c k) ∈ NC )
41	39, 40	mp3an1 1264	. . . . . . . . . . . . . . . 16 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → (2c ↑c k) ∈ NC )
42		tceq 6159	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (m = (2c ↑c k) → Tc m = Tc (2c ↑c k))
43	42	fveq2d 5333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (m = (2c ↑c k) → ( Spac ‘ Tc m) = ( Spac ‘ Tc (2c ↑c k)))
44	43	nceqd 6111	. . . . . . . . . . . . . . . . . . 19 ⊢ (m = (2c ↑c k) → Nc ( Spac ‘ Tc m) = Nc ( Spac ‘ Tc (2c ↑c k)))
45	44	eqeq2d 2364	. . . . . . . . . . . . . . . . . 18 ⊢ (m = (2c ↑c k) → (n = Nc ( Spac ‘ Tc m) ↔ n = Nc ( Spac ‘ Tc (2c ↑c k))))
46		fveq2 5329	. . . . . . . . . . . . . . . . . . . 20 ⊢ (m = (2c ↑c k) → ( Spac ‘m) = ( Spac ‘(2c ↑c k)))
47	46	nceqd 6111	. . . . . . . . . . . . . . . . . . 19 ⊢ (m = (2c ↑c k) → Nc ( Spac ‘m) = Nc ( Spac ‘(2c ↑c k)))
48	47	eleq1d 2419	. . . . . . . . . . . . . . . . . 18 ⊢ (m = (2c ↑c k) → ( Nc ( Spac ‘m) ∈ Nn ↔ Nc ( Spac ‘(2c ↑c k)) ∈ Nn ))
49	45, 48	imbi12d 311	. . . . . . . . . . . . . . . . 17 ⊢ (m = (2c ↑c k) → ((n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )))
50	49	rspcv 2952	. . . . . . . . . . . . . . . 16 ⊢ ((2c ↑c k) ∈ NC → (∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) → (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )))
51	41, 50	syl 15	. . . . . . . . . . . . . . 15 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → (∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) → (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )))
52	51	ancoms 439	. . . . . . . . . . . . . 14 ⊢ (((k ↑c 0c) ∈ NC ∧ k ∈ NC ) → (∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) → (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )))
53	52	adantrl 696	. . . . . . . . . . . . 13 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → (∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) → (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )))
54		tccl 6161	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (k ∈ NC → Tc k ∈ NC )
55		te0c 6238	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (k ∈ NC → ( Tc k ↑c 0c) ∈ NC )
56		nchoicelem7 6296	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (( Tc k ∈ NC ∧ ( Tc k ↑c 0c) ∈ NC ) → Nc ( Spac ‘ Tc k) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c))
57	54, 55, 56	syl2anc 642	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (k ∈ NC → Nc ( Spac ‘ Tc k) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c))
58	57	adantl 452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((n ∈ Nn ∧ k ∈ NC ) → Nc ( Spac ‘ Tc k) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c))
59	58	adantl 452	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → Nc ( Spac ‘ Tc k) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c))
60	59	eqeq2d 2364	. . . . . . . . . . . . . . . . . . 19 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → ((n +c 1c) = Nc ( Spac ‘ Tc k) ↔ (n +c 1c) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c)))
61		nnnc 6147	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (n ∈ Nn → n ∈ NC )
62	61	adantr 451	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((n ∈ Nn ∧ k ∈ NC ) → n ∈ NC )
63	62	adantl 452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → n ∈ NC )
64		fvex 5340	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( Spac ‘(2c ↑c Tc k)) ∈ V
65	64	ncelncsi 6122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Nc ( Spac ‘(2c ↑c Tc k)) ∈ NC
66		peano4nc 6151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((n ∈ NC ∧ Nc ( Spac ‘(2c ↑c Tc k)) ∈ NC ) → ((n +c 1c) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c) ↔ n = Nc ( Spac ‘(2c ↑c Tc k))))
67	63, 65, 66	sylancl 643	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → ((n +c 1c) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c) ↔ n = Nc ( Spac ‘(2c ↑c Tc k))))
68		tce2 6237	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → Tc (2c ↑c k) = (2c ↑c Tc k))
69	68	ancoms 439	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((k ↑c 0c) ∈ NC ∧ k ∈ NC ) → Tc (2c ↑c k) = (2c ↑c Tc k))
70	69	adantrl 696	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → Tc (2c ↑c k) = (2c ↑c Tc k))
71	70	fveq2d 5333	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → ( Spac ‘ Tc (2c ↑c k)) = ( Spac ‘(2c ↑c Tc k)))
72	71	nceqd 6111	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → Nc ( Spac ‘ Tc (2c ↑c k)) = Nc ( Spac ‘(2c ↑c Tc k)))
73	72	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → (n = Nc ( Spac ‘ Tc (2c ↑c k)) ↔ n = Nc ( Spac ‘(2c ↑c Tc k))))
74	73	biimprd 214	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → (n = Nc ( Spac ‘(2c ↑c Tc k)) → n = Nc ( Spac ‘ Tc (2c ↑c k))))
75	67, 74	sylbid 206	. . . . . . . . . . . . . . . . . . 19 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → ((n +c 1c) = ( Nc ( Spac ‘(2c ↑c Tc k)) +c 1c) → n = Nc ( Spac ‘ Tc (2c ↑c k))))
76	60, 75	sylbid 206	. . . . . . . . . . . . . . . . . 18 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → n = Nc ( Spac ‘ Tc (2c ↑c k))))
77	76	imim1d 69	. . . . . . . . . . . . . . . . 17 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → ((n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn ) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )))
78	77	imp 418	. . . . . . . . . . . . . . . 16 ⊢ ((((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) ∧ (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn ))
79		peano2 4404	. . . . . . . . . . . . . . . 16 ⊢ ( Nc ( Spac ‘(2c ↑c k)) ∈ Nn → ( Nc ( Spac ‘(2c ↑c k)) +c 1c) ∈ Nn )
80	78, 79	syl6 29	. . . . . . . . . . . . . . 15 ⊢ ((((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) ∧ (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → ( Nc ( Spac ‘(2c ↑c k)) +c 1c) ∈ Nn ))
81		nchoicelem7 6296	. . . . . . . . . . . . . . . . . . 19 ⊢ ((k ∈ NC ∧ (k ↑c 0c) ∈ NC ) → Nc ( Spac ‘k) = ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
82	81	ancoms 439	. . . . . . . . . . . . . . . . . 18 ⊢ (((k ↑c 0c) ∈ NC ∧ k ∈ NC ) → Nc ( Spac ‘k) = ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
83	82	adantrl 696	. . . . . . . . . . . . . . . . 17 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → Nc ( Spac ‘k) = ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
84	83	adantr 451	. . . . . . . . . . . . . . . 16 ⊢ ((((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) ∧ (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )) → Nc ( Spac ‘k) = ( Nc ( Spac ‘(2c ↑c k)) +c 1c))
85	84	eleq1d 2419	. . . . . . . . . . . . . . 15 ⊢ ((((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) ∧ (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )) → ( Nc ( Spac ‘k) ∈ Nn ↔ ( Nc ( Spac ‘(2c ↑c k)) +c 1c) ∈ Nn ))
86	80, 85	sylibrd 225	. . . . . . . . . . . . . 14 ⊢ ((((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) ∧ (n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn )) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn ))
87	86	ex 423	. . . . . . . . . . . . 13 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → ((n = Nc ( Spac ‘ Tc (2c ↑c k)) → Nc ( Spac ‘(2c ↑c k)) ∈ Nn ) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn )))
88	53, 87	syld 40	. . . . . . . . . . . 12 ⊢ (((k ↑c 0c) ∈ NC ∧ (n ∈ Nn ∧ k ∈ NC )) → (∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn )))
89	88	expimpd 586	. . . . . . . . . . 11 ⊢ ((k ↑c 0c) ∈ NC → (((n ∈ Nn ∧ k ∈ NC ) ∧ ∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn )))
90		nchoicelem3 6292	. . . . . . . . . . . . . . . . 17 ⊢ ((k ∈ NC ∧ ¬ (k ↑c 0c) ∈ NC ) → ( Spac ‘k) = {k})
91	90	nceqd 6111	. . . . . . . . . . . . . . . 16 ⊢ ((k ∈ NC ∧ ¬ (k ↑c 0c) ∈ NC ) → Nc ( Spac ‘k) = Nc {k})
92		vex 2863	. . . . . . . . . . . . . . . . . 18 ⊢ k ∈ V
93	92	df1c3 6141	. . . . . . . . . . . . . . . . 17 ⊢ 1c = Nc {k}
94		1cnnc 4409	. . . . . . . . . . . . . . . . 17 ⊢ 1c ∈ Nn
95	93, 94	eqeltrri 2424	. . . . . . . . . . . . . . . 16 ⊢ Nc {k} ∈ Nn
96	91, 95	syl6eqel 2441	. . . . . . . . . . . . . . 15 ⊢ ((k ∈ NC ∧ ¬ (k ↑c 0c) ∈ NC ) → Nc ( Spac ‘k) ∈ Nn )
97	96	a1d 22	. . . . . . . . . . . . . 14 ⊢ ((k ∈ NC ∧ ¬ (k ↑c 0c) ∈ NC ) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn ))
98	97	expcom 424	. . . . . . . . . . . . 13 ⊢ (¬ (k ↑c 0c) ∈ NC → (k ∈ NC → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn )))
99	98	adantld 453	. . . . . . . . . . . 12 ⊢ (¬ (k ↑c 0c) ∈ NC → ((n ∈ Nn ∧ k ∈ NC ) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn )))
100	99	adantrd 454	. . . . . . . . . . 11 ⊢ (¬ (k ↑c 0c) ∈ NC → (((n ∈ Nn ∧ k ∈ NC ) ∧ ∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn )))
101	89, 100	pm2.61i 156	. . . . . . . . . 10 ⊢ (((n ∈ Nn ∧ k ∈ NC ) ∧ ∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn ))
102	101	an32s 779	. . . . . . . . 9 ⊢ (((n ∈ Nn ∧ ∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )) ∧ k ∈ NC ) → ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn ))
103	102	ralrimiva 2698	. . . . . . . 8 ⊢ ((n ∈ Nn ∧ ∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn )) → ∀k ∈ NC ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn ))
104	103	ex 423	. . . . . . 7 ⊢ (n ∈ Nn → (∀m ∈ NC (n = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) → ∀k ∈ NC ((n +c 1c) = Nc ( Spac ‘ Tc k) → Nc ( Spac ‘k) ∈ Nn )))
105	4, 7, 10, 23, 26, 38, 104	finds 4412	. . . . . 6 ⊢ (x ∈ Nn → ∀m ∈ NC (x = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ))
106		tceq 6159	. . . . . . . . . . 11 ⊢ (m = M → Tc m = Tc M)
107	106	fveq2d 5333	. . . . . . . . . 10 ⊢ (m = M → ( Spac ‘ Tc m) = ( Spac ‘ Tc M))
108	107	nceqd 6111	. . . . . . . . 9 ⊢ (m = M → Nc ( Spac ‘ Tc m) = Nc ( Spac ‘ Tc M))
109	108	eqeq2d 2364	. . . . . . . 8 ⊢ (m = M → (x = Nc ( Spac ‘ Tc m) ↔ x = Nc ( Spac ‘ Tc M)))
110		fveq2 5329	. . . . . . . . . . 11 ⊢ (m = M → ( Spac ‘m) = ( Spac ‘M))
111	110	nceqd 6111	. . . . . . . . . 10 ⊢ (m = M → Nc ( Spac ‘m) = Nc ( Spac ‘M))
112	111	eleq1d 2419	. . . . . . . . 9 ⊢ (m = M → ( Nc ( Spac ‘m) ∈ Nn ↔ Nc ( Spac ‘M) ∈ Nn ))
113		finnc 6244	. . . . . . . . 9 ⊢ (( Spac ‘M) ∈ Fin ↔ Nc ( Spac ‘M) ∈ Nn )
114	112, 113	syl6bbr 254	. . . . . . . 8 ⊢ (m = M → ( Nc ( Spac ‘m) ∈ Nn ↔ ( Spac ‘M) ∈ Fin ))
115	109, 114	imbi12d 311	. . . . . . 7 ⊢ (m = M → ((x = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) ↔ (x = Nc ( Spac ‘ Tc M) → ( Spac ‘M) ∈ Fin )))
116	115	rspccv 2953	. . . . . 6 ⊢ (∀m ∈ NC (x = Nc ( Spac ‘ Tc m) → Nc ( Spac ‘m) ∈ Nn ) → (M ∈ NC → (x = Nc ( Spac ‘ Tc M) → ( Spac ‘M) ∈ Fin )))
117	105, 116	syl 15	. . . . 5 ⊢ (x ∈ Nn → (M ∈ NC → (x = Nc ( Spac ‘ Tc M) → ( Spac ‘M) ∈ Fin )))
118	117	com23 72	. . . 4 ⊢ (x ∈ Nn → (x = Nc ( Spac ‘ Tc M) → (M ∈ NC → ( Spac ‘M) ∈ Fin )))
119	118	rexlimiv 2733	. . 3 ⊢ (∃x ∈ Nn x = Nc ( Spac ‘ Tc M) → (M ∈ NC → ( Spac ‘M) ∈ Fin ))
120	3, 119	sylbi 187	. 2 ⊢ (( Spac ‘ Tc M) ∈ Fin → (M ∈ NC → ( Spac ‘M) ∈ Fin ))
121	120	impcom 419	1 ⊢ ((M ∈ NC ∧ ( Spac ‘ Tc M) ∈ Fin ) → ( Spac ‘M) ∈ Fin )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  ∃wrex 2616  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   Fin cfin 4377   ‘cfv 4782  (class class class)co 5526   NC cncs 6089   Nc cnc 6092   T_c ctc 6094  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-lec 6100  df-ltc 6101  df-nc 6102  df-tc 6104  df-2c 6105  df-ce 6107  df-tcfn 6108  df-spac 6275

This theorem is used by:  nchoicelem19  6308