Theorem nchoicelem9 6298

Description: Lemma for nchoice 6309. Calculate the cardinality of the special set generator when near the end of raisability. Theorem 6.8 of [Specker] p. 974. (Contributed by SF, 18-Mar-2015.)

Assertion

Ref	Expression
nchoicelem9	⊢ (( ≤c We NC ∧ M ∈ NC ∧ ¬ (M ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘ Tc M) = 2c ∨ Nc ( Spac ‘ Tc M) = 3c))

Proof of Theorem nchoicelem9

Step	Hyp	Ref	Expression
1		brltc 6115	. . . . 5 ⊢ ( Nc 1c <c M ↔ ( Nc 1c ≤c M ∧ Nc 1c ≠ M))
2	1	simplbi 446	. . . 4 ⊢ ( Nc 1c <c M → Nc 1c ≤c M)
3		1cex 4143	. . . . . . . 8 ⊢ 1c ∈ V
4	3	ncelncsi 6122	. . . . . . 7 ⊢ Nc 1c ∈ NC
5		tlecg 6231	. . . . . . 7 ⊢ (( Nc 1c ∈ NC ∧ M ∈ NC ) → ( Nc 1c ≤c M ↔ Tc Nc 1c ≤c Tc M))
6	4, 5	mpan 651	. . . . . 6 ⊢ (M ∈ NC → ( Nc 1c ≤c M ↔ Tc Nc 1c ≤c Tc M))
7	6	adantl 452	. . . . 5 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Nc 1c ≤c M ↔ Tc Nc 1c ≤c Tc M))
8		tcnc1c 6228	. . . . . . 7 ⊢ Tc Nc 1c = Nc 𝒫11c
9	8	breq1i 4647	. . . . . 6 ⊢ ( Tc Nc 1c ≤c Tc M ↔ Nc 𝒫11c ≤c Tc M)
10		tccl 6161	. . . . . . . . 9 ⊢ (M ∈ NC → Tc M ∈ NC )
11		te0c 6238	. . . . . . . . 9 ⊢ (M ∈ NC → ( Tc M ↑c 0c) ∈ NC )
12	3	pw1ex 4304	. . . . . . . . . . 11 ⊢ 𝒫11c ∈ V
13	12	ncelncsi 6122	. . . . . . . . . 10 ⊢ Nc 𝒫11c ∈ NC
14		ce2le 6234	. . . . . . . . . . 11 ⊢ ((( Nc 𝒫11c ∈ NC ∧ Tc M ∈ NC ∧ ( Tc M ↑c 0c) ∈ NC ) ∧ Nc 𝒫11c ≤c Tc M) → (2c ↑c Nc 𝒫11c) ≤c (2c ↑c Tc M))
15	14	ex 423	. . . . . . . . . 10 ⊢ (( Nc 𝒫11c ∈ NC ∧ Tc M ∈ NC ∧ ( Tc M ↑c 0c) ∈ NC ) → ( Nc 𝒫11c ≤c Tc M → (2c ↑c Nc 𝒫11c) ≤c (2c ↑c Tc M)))
16	13, 15	mp3an1 1264	. . . . . . . . 9 ⊢ (( Tc M ∈ NC ∧ ( Tc M ↑c 0c) ∈ NC ) → ( Nc 𝒫11c ≤c Tc M → (2c ↑c Nc 𝒫11c) ≤c (2c ↑c Tc M)))
17	10, 11, 16	syl2anc 642	. . . . . . . 8 ⊢ (M ∈ NC → ( Nc 𝒫11c ≤c Tc M → (2c ↑c Nc 𝒫11c) ≤c (2c ↑c Tc M)))
18	17	adantl 452	. . . . . . 7 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Nc 𝒫11c ≤c Tc M → (2c ↑c Nc 𝒫11c) ≤c (2c ↑c Tc M)))
19		ce2ncpw11c 6195	. . . . . . . . 9 ⊢ (2c ↑c Nc 𝒫11c) = Nc 1c
20	19	breq1i 4647	. . . . . . . 8 ⊢ ((2c ↑c Nc 𝒫11c) ≤c (2c ↑c Tc M) ↔ Nc 1c ≤c (2c ↑c Tc M))
21		orc 374	. . . . . . . . . 10 ⊢ ( Nc 1c ≤c (2c ↑c Tc M) → ( Nc 1c ≤c (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)))
22		brltc 6115	. . . . . . . . . . . 12 ⊢ ( Nc 1c <c (2c ↑c Tc M) ↔ ( Nc 1c ≤c (2c ↑c Tc M) ∧ Nc 1c ≠ (2c ↑c Tc M)))
23	22	orbi1i 506	. . . . . . . . . . 11 ⊢ (( Nc 1c <c (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)) ↔ (( Nc 1c ≤c (2c ↑c Tc M) ∧ Nc 1c ≠ (2c ↑c Tc M)) ∨ Nc 1c = (2c ↑c Tc M)))
24		pm2.1 406	. . . . . . . . . . . . 13 ⊢ (¬ Nc 1c = (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M))
25		df-ne 2519	. . . . . . . . . . . . . 14 ⊢ ( Nc 1c ≠ (2c ↑c Tc M) ↔ ¬ Nc 1c = (2c ↑c Tc M))
26	25	orbi1i 506	. . . . . . . . . . . . 13 ⊢ (( Nc 1c ≠ (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)) ↔ (¬ Nc 1c = (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)))
27	24, 26	mpbir 200	. . . . . . . . . . . 12 ⊢ ( Nc 1c ≠ (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M))
28		ordir 835	. . . . . . . . . . . 12 ⊢ ((( Nc 1c ≤c (2c ↑c Tc M) ∧ Nc 1c ≠ (2c ↑c Tc M)) ∨ Nc 1c = (2c ↑c Tc M)) ↔ (( Nc 1c ≤c (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)) ∧ ( Nc 1c ≠ (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M))))
29	27, 28	mpbiran2 885	. . . . . . . . . . 11 ⊢ ((( Nc 1c ≤c (2c ↑c Tc M) ∧ Nc 1c ≠ (2c ↑c Tc M)) ∨ Nc 1c = (2c ↑c Tc M)) ↔ ( Nc 1c ≤c (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)))
30	23, 29	bitri 240	. . . . . . . . . 10 ⊢ (( Nc 1c <c (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)) ↔ ( Nc 1c ≤c (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)))
31	21, 30	sylibr 203	. . . . . . . . 9 ⊢ ( Nc 1c ≤c (2c ↑c Tc M) → ( Nc 1c <c (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)))
32		ce2t 6236	. . . . . . . . . . . 12 ⊢ (M ∈ NC → (2c ↑c Tc M) ∈ NC )
33		nchoicelem8 6297	. . . . . . . . . . . 12 ⊢ (( ≤c We NC ∧ (2c ↑c Tc M) ∈ NC ) → (¬ ((2c ↑c Tc M) ↑c 0c) ∈ NC ↔ Nc 1c <c (2c ↑c Tc M)))
34	32, 33	sylan2 460	. . . . . . . . . . 11 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ ((2c ↑c Tc M) ↑c 0c) ∈ NC ↔ Nc 1c <c (2c ↑c Tc M)))
35		nchoicelem3 6292	. . . . . . . . . . . . . . . 16 ⊢ (((2c ↑c Tc M) ∈ NC ∧ ¬ ((2c ↑c Tc M) ↑c 0c) ∈ NC ) → ( Spac ‘(2c ↑c Tc M)) = {(2c ↑c Tc M)})
36	35	nceqd 6111	. . . . . . . . . . . . . . 15 ⊢ (((2c ↑c Tc M) ∈ NC ∧ ¬ ((2c ↑c Tc M) ↑c 0c) ∈ NC ) → Nc ( Spac ‘(2c ↑c Tc M)) = Nc {(2c ↑c Tc M)})
37		ovex 5552	. . . . . . . . . . . . . . . 16 ⊢ (2c ↑c Tc M) ∈ V
38	37	df1c3 6141	. . . . . . . . . . . . . . 15 ⊢ 1c = Nc {(2c ↑c Tc M)}
39	36, 38	syl6eqr 2403	. . . . . . . . . . . . . 14 ⊢ (((2c ↑c Tc M) ∈ NC ∧ ¬ ((2c ↑c Tc M) ↑c 0c) ∈ NC ) → Nc ( Spac ‘(2c ↑c Tc M)) = 1c)
40	39	ex 423	. . . . . . . . . . . . 13 ⊢ ((2c ↑c Tc M) ∈ NC → (¬ ((2c ↑c Tc M) ↑c 0c) ∈ NC → Nc ( Spac ‘(2c ↑c Tc M)) = 1c))
41	32, 40	syl 15	. . . . . . . . . . . 12 ⊢ (M ∈ NC → (¬ ((2c ↑c Tc M) ↑c 0c) ∈ NC → Nc ( Spac ‘(2c ↑c Tc M)) = 1c))
42	41	adantl 452	. . . . . . . . . . 11 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ ((2c ↑c Tc M) ↑c 0c) ∈ NC → Nc ( Spac ‘(2c ↑c Tc M)) = 1c))
43	34, 42	sylbird 226	. . . . . . . . . 10 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Nc 1c <c (2c ↑c Tc M) → Nc ( Spac ‘(2c ↑c Tc M)) = 1c))
44		nclecid 6198	. . . . . . . . . . . . . . . . . . 19 ⊢ ( Nc 1c ∈ NC → Nc 1c ≤c Nc 1c)
45	4, 44	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ Nc 1c ≤c Nc 1c
46		ce0lenc1 6240	. . . . . . . . . . . . . . . . . . 19 ⊢ ( Nc 1c ∈ NC → (( Nc 1c ↑c 0c) ∈ NC ↔ Nc 1c ≤c Nc 1c))
47	4, 46	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (( Nc 1c ↑c 0c) ∈ NC ↔ Nc 1c ≤c Nc 1c)
48	45, 47	mpbir 200	. . . . . . . . . . . . . . . . 17 ⊢ ( Nc 1c ↑c 0c) ∈ NC
49		ce2lt 6221	. . . . . . . . . . . . . . . . 17 ⊢ (( Nc 1c ∈ NC ∧ ( Nc 1c ↑c 0c) ∈ NC ) → Nc 1c <c (2c ↑c Nc 1c))
50	4, 48, 49	mp2an 653	. . . . . . . . . . . . . . . 16 ⊢ Nc 1c <c (2c ↑c Nc 1c)
51		2nnc 6168	. . . . . . . . . . . . . . . . . 18 ⊢ 2c ∈ Nn
52		ceclnn1 6190	. . . . . . . . . . . . . . . . . 18 ⊢ ((2c ∈ Nn ∧ Nc 1c ∈ NC ∧ ( Nc 1c ↑c 0c) ∈ NC ) → (2c ↑c Nc 1c) ∈ NC )
53	51, 4, 48, 52	mp3an 1277	. . . . . . . . . . . . . . . . 17 ⊢ (2c ↑c Nc 1c) ∈ NC
54		nchoicelem8 6297	. . . . . . . . . . . . . . . . 17 ⊢ (( ≤c We NC ∧ (2c ↑c Nc 1c) ∈ NC ) → (¬ ((2c ↑c Nc 1c) ↑c 0c) ∈ NC ↔ Nc 1c <c (2c ↑c Nc 1c)))
55	53, 54	mpan2 652	. . . . . . . . . . . . . . . 16 ⊢ ( ≤c We NC → (¬ ((2c ↑c Nc 1c) ↑c 0c) ∈ NC ↔ Nc 1c <c (2c ↑c Nc 1c)))
56	50, 55	mpbiri 224	. . . . . . . . . . . . . . 15 ⊢ ( ≤c We NC → ¬ ((2c ↑c Nc 1c) ↑c 0c) ∈ NC )
57		nchoicelem3 6292	. . . . . . . . . . . . . . . . . 18 ⊢ (((2c ↑c Nc 1c) ∈ NC ∧ ¬ ((2c ↑c Nc 1c) ↑c 0c) ∈ NC ) → ( Spac ‘(2c ↑c Nc 1c)) = {(2c ↑c Nc 1c)})
58	57	nceqd 6111	. . . . . . . . . . . . . . . . 17 ⊢ (((2c ↑c Nc 1c) ∈ NC ∧ ¬ ((2c ↑c Nc 1c) ↑c 0c) ∈ NC ) → Nc ( Spac ‘(2c ↑c Nc 1c)) = Nc {(2c ↑c Nc 1c)})
59		ovex 5552	. . . . . . . . . . . . . . . . . 18 ⊢ (2c ↑c Nc 1c) ∈ V
60	59	df1c3 6141	. . . . . . . . . . . . . . . . 17 ⊢ 1c = Nc {(2c ↑c Nc 1c)}
61	58, 60	syl6eqr 2403	. . . . . . . . . . . . . . . 16 ⊢ (((2c ↑c Nc 1c) ∈ NC ∧ ¬ ((2c ↑c Nc 1c) ↑c 0c) ∈ NC ) → Nc ( Spac ‘(2c ↑c Nc 1c)) = 1c)
62	53, 61	mpan 651	. . . . . . . . . . . . . . 15 ⊢ (¬ ((2c ↑c Nc 1c) ↑c 0c) ∈ NC → Nc ( Spac ‘(2c ↑c Nc 1c)) = 1c)
63	56, 62	syl 15	. . . . . . . . . . . . . 14 ⊢ ( ≤c We NC → Nc ( Spac ‘(2c ↑c Nc 1c)) = 1c)
64	63	addceq1d 4390	. . . . . . . . . . . . 13 ⊢ ( ≤c We NC → ( Nc ( Spac ‘(2c ↑c Nc 1c)) +c 1c) = (1c +c 1c))
65		nchoicelem7 6296	. . . . . . . . . . . . . 14 ⊢ (( Nc 1c ∈ NC ∧ ( Nc 1c ↑c 0c) ∈ NC ) → Nc ( Spac ‘ Nc 1c) = ( Nc ( Spac ‘(2c ↑c Nc 1c)) +c 1c))
66	4, 48, 65	mp2an 653	. . . . . . . . . . . . 13 ⊢ Nc ( Spac ‘ Nc 1c) = ( Nc ( Spac ‘(2c ↑c Nc 1c)) +c 1c)
67		1p1e2c 6156	. . . . . . . . . . . . . 14 ⊢ (1c +c 1c) = 2c
68	67	eqcomi 2357	. . . . . . . . . . . . 13 ⊢ 2c = (1c +c 1c)
69	64, 66, 68	3eqtr4g 2410	. . . . . . . . . . . 12 ⊢ ( ≤c We NC → Nc ( Spac ‘ Nc 1c) = 2c)
70		fveq2 5329	. . . . . . . . . . . . . 14 ⊢ ( Nc 1c = (2c ↑c Tc M) → ( Spac ‘ Nc 1c) = ( Spac ‘(2c ↑c Tc M)))
71	70	nceqd 6111	. . . . . . . . . . . . 13 ⊢ ( Nc 1c = (2c ↑c Tc M) → Nc ( Spac ‘ Nc 1c) = Nc ( Spac ‘(2c ↑c Tc M)))
72	71	eqeq1d 2361	. . . . . . . . . . . 12 ⊢ ( Nc 1c = (2c ↑c Tc M) → ( Nc ( Spac ‘ Nc 1c) = 2c ↔ Nc ( Spac ‘(2c ↑c Tc M)) = 2c))
73	69, 72	syl5ibcom 211	. . . . . . . . . . 11 ⊢ ( ≤c We NC → ( Nc 1c = (2c ↑c Tc M) → Nc ( Spac ‘(2c ↑c Tc M)) = 2c))
74	73	adantr 451	. . . . . . . . . 10 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Nc 1c = (2c ↑c Tc M) → Nc ( Spac ‘(2c ↑c Tc M)) = 2c))
75	43, 74	orim12d 811	. . . . . . . . 9 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (( Nc 1c <c (2c ↑c Tc M) ∨ Nc 1c = (2c ↑c Tc M)) → ( Nc ( Spac ‘(2c ↑c Tc M)) = 1c ∨ Nc ( Spac ‘(2c ↑c Tc M)) = 2c)))
76	31, 75	syl5 28	. . . . . . . 8 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Nc 1c ≤c (2c ↑c Tc M) → ( Nc ( Spac ‘(2c ↑c Tc M)) = 1c ∨ Nc ( Spac ‘(2c ↑c Tc M)) = 2c)))
77	20, 76	syl5bi 208	. . . . . . 7 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ((2c ↑c Nc 𝒫11c) ≤c (2c ↑c Tc M) → ( Nc ( Spac ‘(2c ↑c Tc M)) = 1c ∨ Nc ( Spac ‘(2c ↑c Tc M)) = 2c)))
78	18, 77	syld 40	. . . . . 6 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Nc 𝒫11c ≤c Tc M → ( Nc ( Spac ‘(2c ↑c Tc M)) = 1c ∨ Nc ( Spac ‘(2c ↑c Tc M)) = 2c)))
79	9, 78	syl5bi 208	. . . . 5 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Tc Nc 1c ≤c Tc M → ( Nc ( Spac ‘(2c ↑c Tc M)) = 1c ∨ Nc ( Spac ‘(2c ↑c Tc M)) = 2c)))
80	7, 79	sylbid 206	. . . 4 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Nc 1c ≤c M → ( Nc ( Spac ‘(2c ↑c Tc M)) = 1c ∨ Nc ( Spac ‘(2c ↑c Tc M)) = 2c)))
81		addceq1 4384	. . . . 5 ⊢ ( Nc ( Spac ‘(2c ↑c Tc M)) = 1c → ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (1c +c 1c))
82		addceq1 4384	. . . . 5 ⊢ ( Nc ( Spac ‘(2c ↑c Tc M)) = 2c → ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (2c +c 1c))
83	81, 82	orim12i 502	. . . 4 ⊢ (( Nc ( Spac ‘(2c ↑c Tc M)) = 1c ∨ Nc ( Spac ‘(2c ↑c Tc M)) = 2c) → (( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (1c +c 1c) ∨ ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (2c +c 1c)))
84	2, 80, 83	syl56 30	. . 3 ⊢ (( ≤c We NC ∧ M ∈ NC ) → ( Nc 1c <c M → (( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (1c +c 1c) ∨ ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (2c +c 1c))))
85		nchoicelem8 6297	. . 3 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ (M ↑c 0c) ∈ NC ↔ Nc 1c <c M))
86		nchoicelem7 6296	. . . . . . 7 ⊢ (( Tc M ∈ NC ∧ ( Tc M ↑c 0c) ∈ NC ) → Nc ( Spac ‘ Tc M) = ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c))
87	10, 11, 86	syl2anc 642	. . . . . 6 ⊢ (M ∈ NC → Nc ( Spac ‘ Tc M) = ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c))
88	68	a1i 10	. . . . . 6 ⊢ (M ∈ NC → 2c = (1c +c 1c))
89	87, 88	eqeq12d 2367	. . . . 5 ⊢ (M ∈ NC → ( Nc ( Spac ‘ Tc M) = 2c ↔ ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (1c +c 1c)))
90		2p1e3c 6157	. . . . . . . 8 ⊢ (2c +c 1c) = 3c
91	90	eqcomi 2357	. . . . . . 7 ⊢ 3c = (2c +c 1c)
92	91	a1i 10	. . . . . 6 ⊢ (M ∈ NC → 3c = (2c +c 1c))
93	87, 92	eqeq12d 2367	. . . . 5 ⊢ (M ∈ NC → ( Nc ( Spac ‘ Tc M) = 3c ↔ ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (2c +c 1c)))
94	89, 93	orbi12d 690	. . . 4 ⊢ (M ∈ NC → (( Nc ( Spac ‘ Tc M) = 2c ∨ Nc ( Spac ‘ Tc M) = 3c) ↔ (( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (1c +c 1c) ∨ ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (2c +c 1c))))
95	94	adantl 452	. . 3 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (( Nc ( Spac ‘ Tc M) = 2c ∨ Nc ( Spac ‘ Tc M) = 3c) ↔ (( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (1c +c 1c) ∨ ( Nc ( Spac ‘(2c ↑c Tc M)) +c 1c) = (2c +c 1c))))
96	84, 85, 95	3imtr4d 259	. 2 ⊢ (( ≤c We NC ∧ M ∈ NC ) → (¬ (M ↑c 0c) ∈ NC → ( Nc ( Spac ‘ Tc M) = 2c ∨ Nc ( Spac ‘ Tc M) = 3c)))
97	96	3impia 1148	1 ⊢ (( ≤c We NC ∧ M ∈ NC ∧ ¬ (M ↑c 0c) ∈ NC ) → ( Nc ( Spac ‘ Tc M) = 2c ∨ Nc ( Spac ‘ Tc M) = 3c))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  {csn 3738  1_cc1c 4135  𝒫₁cpw1 4136   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   class class class wbr 4640   ‘cfv 4782  (class class class)co 5526   We cwe 5896   NC cncs 6089   ≤_c clec 6090   <_c cltc 6091   Nc cnc 6092   T_c ctc 6094  2_cc2c 6095  3_cc3c 6096   ↑_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-tp 3744  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-antisym 5902  df-partial 5903  df-connex 5904  df-strict 5905  df-we 5907  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-3c 6106  df-ce 6107  df-spac 6275

This theorem is used by:  nchoicelem17  6306