Theorem exmidonfinlem 7149

Description: Lemma for exmidonfin 7150. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)

Hypothesis

Ref	Expression
exmidonfinlem.a	⊢ 𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}

Assertion

Ref	Expression
exmidonfinlem	⊢ (ω = (On ∩ Fin) → DECID 𝜑)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem exmidonfinlem

Dummy variables 𝑟 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpri 3599	. . . . . . . . . 10 ⊢ (𝑟 ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
2		exmidonfinlem.a	. . . . . . . . . 10 ⊢ 𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}
3	1, 2	eleq2s 2261	. . . . . . . . 9 ⊢ (𝑟 ∈ 𝐴 → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
4		eleq2 2230	. . . . . . . . . . . 12 ⊢ (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} → (𝑠 ∈ 𝑟 ↔ 𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
5	4	biimpcd 158	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
6		elrabi 2879	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ {∅})
7		velsn 3593	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {∅} ↔ 𝑠 = ∅)
8	6, 7	sylib 121	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 = ∅)
9		biidd 171	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑠 → (𝜑 ↔ 𝜑))
10	9	elrab 2882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ (𝑠 ∈ {∅} ∧ 𝜑))
11	10	simprbi 273	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝜑)
12	11	notnotd 620	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ ¬ 𝜑)
13		0ex 4109	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
14	13	snm 3696	. . . . . . . . . . . . . . . 16 ⊢ ∃𝑤 𝑤 ∈ {∅}
15		r19.3rmv 3499	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
16	14, 15	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
17	12, 16	sylib 121	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
18		rabeq0 3438	. . . . . . . . . . . . . 14 ⊢ ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
19	17, 18	sylibr 133	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
20	8, 19	eqtr4d 2201	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 = {𝑥 ∈ {∅} ∣ ¬ 𝜑})
21		p0ex 4167	. . . . . . . . . . . . . . 15 ⊢ {∅} ∈ V
22	21	rabex 4126	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ V
23	22	prid2 3683	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}
24	23, 2	eleqtrri 2242	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ 𝐴
25	20, 24	eqeltrdi 2257	. . . . . . . . . . 11 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ 𝐴)
26	5, 25	syl6 33	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ 𝐴))
27		eleq2 2230	. . . . . . . . . . . 12 ⊢ (𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (𝑠 ∈ 𝑟 ↔ 𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
28	27	biimpcd 158	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
29		elrabi 2879	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ {∅})
30	29, 7	sylib 121	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 = ∅)
31		biidd 171	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑠 → (¬ 𝜑 ↔ ¬ 𝜑))
32	31	elrab 2882	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑠 ∈ {∅} ∧ ¬ 𝜑))
33	32	simprbi 273	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
34		r19.3rmv 3499	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 𝑤 ∈ {∅} → (¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ 𝜑))
35	14, 34	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ 𝜑)
36	33, 35	sylib 121	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ∀𝑥 ∈ {∅} ¬ 𝜑)
37		rabeq0 3438	. . . . . . . . . . . . . 14 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ 𝜑)
38	36, 37	sylibr 133	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → {𝑥 ∈ {∅} ∣ 𝜑} = ∅)
39	30, 38	eqtr4d 2201	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 = {𝑥 ∈ {∅} ∣ 𝜑})
40	21	rabex 4126	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
41	40	prid1 3682	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}
42	41, 2	eleqtrri 2242	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ 𝐴
43	39, 42	eqeltrdi 2257	. . . . . . . . . . 11 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ 𝐴)
44	28, 43	syl6 33	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ 𝐴))
45	26, 44	jaod 707	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝑟 → ((𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → 𝑠 ∈ 𝐴))
46	3, 45	mpan9 279	. . . . . . . 8 ⊢ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ 𝐴)
47	46	rgen2 2552	. . . . . . 7 ⊢ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝑟 𝑠 ∈ 𝐴
48		dftr5 4083	. . . . . . 7 ⊢ (Tr 𝐴 ↔ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝑟 𝑠 ∈ 𝐴)
49	47, 48	mpbir 145	. . . . . 6 ⊢ Tr 𝐴
50		elpri 3599	. . . . . . . . 9 ⊢ (𝑧 ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} → (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
51	50, 2	eleq2s 2261	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
52		ordtriexmidlem 4496	. . . . . . . . . . 11 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On
53	52	ontrci 4405	. . . . . . . . . 10 ⊢ Tr {𝑥 ∈ {∅} ∣ 𝜑}
54		treq 4086	. . . . . . . . . 10 ⊢ (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} → (Tr 𝑧 ↔ Tr {𝑥 ∈ {∅} ∣ 𝜑}))
55	53, 54	mpbiri 167	. . . . . . . . 9 ⊢ (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} → Tr 𝑧)
56		ordtriexmidlem 4496	. . . . . . . . . . 11 ⊢ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ On
57	56	ontrci 4405	. . . . . . . . . 10 ⊢ Tr {𝑥 ∈ {∅} ∣ ¬ 𝜑}
58		treq 4086	. . . . . . . . . 10 ⊢ (𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (Tr 𝑧 ↔ Tr {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
59	57, 58	mpbiri 167	. . . . . . . . 9 ⊢ (𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → Tr 𝑧)
60	55, 59	jaoi 706	. . . . . . . 8 ⊢ ((𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → Tr 𝑧)
61	51, 60	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → Tr 𝑧)
62	61	rgen 2519	. . . . . 6 ⊢ ∀𝑧 ∈ 𝐴 Tr 𝑧
63		dford3 4345	. . . . . 6 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 Tr 𝑧))
64	49, 62, 63	mpbir2an 932	. . . . 5 ⊢ Ord 𝐴
65		prexg 4189	. . . . . . . 8 ⊢ (({𝑥 ∈ {∅} ∣ 𝜑} ∈ V ∧ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ V) → {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ∈ V)
66	40, 22, 65	mp2an 423	. . . . . . 7 ⊢ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ∈ V
67	2, 66	eqeltri 2239	. . . . . 6 ⊢ 𝐴 ∈ V
68	67	elon 4352	. . . . 5 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)
69	64, 68	mpbir 145	. . . 4 ⊢ 𝐴 ∈ On
70		2onn 6489	. . . . . 6 ⊢ 2o ∈ ω
71		nnfi 6838	. . . . . 6 ⊢ (2o ∈ ω → 2o ∈ Fin)
72	70, 71	ax-mp 5	. . . . 5 ⊢ 2o ∈ Fin
73		pm5.19 696	. . . . . . . . . 10 ⊢ ¬ (𝜑 ↔ ¬ 𝜑)
74	13	snm 3696	. . . . . . . . . . 11 ⊢ ∃𝑦 𝑦 ∈ {∅}
75		r19.3rmv 3499	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑦 ∈ {∅} → ((𝜑 ↔ ¬ 𝜑) ↔ ∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑)))
76	74, 75	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝜑 ↔ ¬ 𝜑) ↔ ∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑))
77	73, 76	mtbi 660	. . . . . . . . 9 ⊢ ¬ ∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑)
78		rabbi 2643	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑) ↔ {𝑥 ∈ {∅} ∣ 𝜑} = {𝑥 ∈ {∅} ∣ ¬ 𝜑})
79	77, 78	mtbi 660	. . . . . . . 8 ⊢ ¬ {𝑥 ∈ {∅} ∣ 𝜑} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
80	79	neir 2339	. . . . . . 7 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ≠ {𝑥 ∈ {∅} ∣ ¬ 𝜑}
81		pr2ne 7148	. . . . . . . 8 ⊢ (({𝑥 ∈ {∅} ∣ 𝜑} ∈ V ∧ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ V) → ({{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ≈ 2o ↔ {𝑥 ∈ {∅} ∣ 𝜑} ≠ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
82	40, 22, 81	mp2an 423	. . . . . . 7 ⊢ ({{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ≈ 2o ↔ {𝑥 ∈ {∅} ∣ 𝜑} ≠ {𝑥 ∈ {∅} ∣ ¬ 𝜑})
83	80, 82	mpbir 145	. . . . . 6 ⊢ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ≈ 2o
84	2, 83	eqbrtri 4003	. . . . 5 ⊢ 𝐴 ≈ 2o
85		enfii 6840	. . . . 5 ⊢ ((2o ∈ Fin ∧ 𝐴 ≈ 2o) → 𝐴 ∈ Fin)
86	72, 84, 85	mp2an 423	. . . 4 ⊢ 𝐴 ∈ Fin
87	69, 86	elini 3306	. . 3 ⊢ 𝐴 ∈ (On ∩ Fin)
88		eleq2 2230	. . 3 ⊢ (ω = (On ∩ Fin) → (𝐴 ∈ ω ↔ 𝐴 ∈ (On ∩ Fin)))
89	87, 88	mpbiri 167	. 2 ⊢ (ω = (On ∩ Fin) → 𝐴 ∈ ω)
90		df1o2 6397	. . . . 5 ⊢ 1o = {∅}
91		1lt2o 6410	. . . . 5 ⊢ 1o ∈ 2o
92	90, 91	eqeltrri 2240	. . . 4 ⊢ {∅} ∈ 2o
93		nneneq 6823	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 2o ∈ ω) → (𝐴 ≈ 2o ↔ 𝐴 = 2o))
94	70, 93	mpan2 422	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ≈ 2o ↔ 𝐴 = 2o))
95	84, 94	mpbii 147	. . . 4 ⊢ (𝐴 ∈ ω → 𝐴 = 2o)
96	92, 95	eleqtrrid 2256	. . 3 ⊢ (𝐴 ∈ ω → {∅} ∈ 𝐴)
97		elpri 3599	. . . 4 ⊢ ({∅} ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} → ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
98	97, 2	eleq2s 2261	. . 3 ⊢ ({∅} ∈ 𝐴 → ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
99	96, 98	syl 14	. 2 ⊢ (𝐴 ∈ ω → ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
100	13	snid 3607	. . . . . . 7 ⊢ ∅ ∈ {∅}
101		eleq2 2230	. . . . . . 7 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → (∅ ∈ {∅} ↔ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
102	100, 101	mpbii 147	. . . . . 6 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
103		biidd 171	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑))
104	103	elrab 2882	. . . . . 6 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {∅} ∧ 𝜑))
105	102, 104	sylib 121	. . . . 5 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → (∅ ∈ {∅} ∧ 𝜑))
106	105	simprd 113	. . . 4 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → 𝜑)
107		eleq2 2230	. . . . . . 7 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (∅ ∈ {∅} ↔ ∅ ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
108	100, 107	mpbii 147	. . . . . 6 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ∅ ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑})
109		biidd 171	. . . . . . 7 ⊢ (𝑥 = ∅ → (¬ 𝜑 ↔ ¬ 𝜑))
110	109	elrab 2882	. . . . . 6 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (∅ ∈ {∅} ∧ ¬ 𝜑))
111	108, 110	sylib 121	. . . . 5 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (∅ ∈ {∅} ∧ ¬ 𝜑))
112	111	simprd 113	. . . 4 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
113	106, 112	orim12i 749	. . 3 ⊢ (({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
114		df-dc 825	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
115	113, 114	sylibr 133	. 2 ⊢ (({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → DECID 𝜑)
116	89, 99, 115	3syl 17	1 ⊢ (ω = (On ∩ Fin) → DECID 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 824   = wceq 1343  ∃wex 1480   ∈ wcel 2136   ≠ wne 2336  ∀wral 2444  {crab 2448  Vcvv 2726   ∩ cin 3115  ∅c0 3409  {csn 3576  {cpr 3577   class class class wbr 3982  Tr wtr 4080  Ord word 4340  Oncon0 4341  ωcom 4567  1_oc1o 6377  2_oc2o 6378   ≈ cen 6704  Fincfn 6706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1o 6384  df-2o 6385  df-er 6501  df-en 6707  df-fin 6709

This theorem is referenced by:  exmidonfin  7150