Theorem exmidonfinlem 7496

Description: Lemma for exmidonfin 7497. (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 3712	. . . . . . . . . 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 2327	. . . . . . . . 9 ⊢ (𝑟 ∈ 𝐴 → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
4		eleq2 2296	. . . . . . . . . . . 12 ⊢ (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} → (𝑠 ∈ 𝑟 ↔ 𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
5	4	biimpcd 159	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
6		elrabi 2970	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ {∅})
7		velsn 3706	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {∅} ↔ 𝑠 = ∅)
8	6, 7	sylib 122	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 = ∅)
9		biidd 172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑠 → (𝜑 ↔ 𝜑))
10	9	elrab 2973	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ (𝑠 ∈ {∅} ∧ 𝜑))
11	10	simprbi 275	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝜑)
12	11	notnotd 635	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ ¬ 𝜑)
13		0ex 4237	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
14	13	snm 3812	. . . . . . . . . . . . . . . 16 ⊢ ∃𝑤 𝑤 ∈ {∅}
15		r19.3rmv 3600	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
16	14, 15	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
17	12, 16	sylib 122	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
18		rabeq0 3538	. . . . . . . . . . . . . 14 ⊢ ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
19	17, 18	sylibr 134	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
20	8, 19	eqtr4d 2268	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 = {𝑥 ∈ {∅} ∣ ¬ 𝜑})
21		p0ex 4301	. . . . . . . . . . . . . . 15 ⊢ {∅} ∈ V
22	21	rabex 4256	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ V
23	22	prid2 3798	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}
24	23, 2	eleqtrri 2308	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ 𝐴
25	20, 24	eqeltrdi 2323	. . . . . . . . . . 11 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ 𝐴)
26	5, 25	syl6 33	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ 𝐴))
27		eleq2 2296	. . . . . . . . . . . 12 ⊢ (𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (𝑠 ∈ 𝑟 ↔ 𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
28	27	biimpcd 159	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
29		elrabi 2970	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ {∅})
30	29, 7	sylib 122	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 = ∅)
31		biidd 172	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑠 → (¬ 𝜑 ↔ ¬ 𝜑))
32	31	elrab 2973	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑠 ∈ {∅} ∧ ¬ 𝜑))
33	32	simprbi 275	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
34		r19.3rmv 3600	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤 𝑤 ∈ {∅} → (¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ 𝜑))
35	14, 34	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ 𝜑)
36	33, 35	sylib 122	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ∀𝑥 ∈ {∅} ¬ 𝜑)
37		rabeq0 3538	. . . . . . . . . . . . . 14 ⊢ ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ 𝜑)
38	36, 37	sylibr 134	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → {𝑥 ∈ {∅} ∣ 𝜑} = ∅)
39	30, 38	eqtr4d 2268	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 = {𝑥 ∈ {∅} ∣ 𝜑})
40	21	rabex 4256	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
41	40	prid1 3797	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}
42	41, 2	eleqtrri 2308	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ 𝐴
43	39, 42	eqeltrdi 2323	. . . . . . . . . . 11 ⊢ (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ 𝐴)
44	28, 43	syl6 33	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ 𝐴))
45	26, 44	jaod 725	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝑟 → ((𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → 𝑠 ∈ 𝐴))
46	3, 45	mpan9 281	. . . . . . . 8 ⊢ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝑟) → 𝑠 ∈ 𝐴)
47	46	rgen2 2628	. . . . . . 7 ⊢ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝑟 𝑠 ∈ 𝐴
48		dftr5 4211	. . . . . . 7 ⊢ (Tr 𝐴 ↔ ∀𝑟 ∈ 𝐴 ∀𝑠 ∈ 𝑟 𝑠 ∈ 𝐴)
49	47, 48	mpbir 146	. . . . . 6 ⊢ Tr 𝐴
50		elpri 3712	. . . . . . . . 9 ⊢ (𝑧 ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} → (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
51	50, 2	eleq2s 2327	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
52		ordtriexmidlem 4641	. . . . . . . . . . 11 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ∈ On
53	52	ontrci 4548	. . . . . . . . . 10 ⊢ Tr {𝑥 ∈ {∅} ∣ 𝜑}
54		treq 4214	. . . . . . . . . 10 ⊢ (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} → (Tr 𝑧 ↔ Tr {𝑥 ∈ {∅} ∣ 𝜑}))
55	53, 54	mpbiri 168	. . . . . . . . 9 ⊢ (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} → Tr 𝑧)
56		ordtriexmidlem 4641	. . . . . . . . . . 11 ⊢ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ On
57	56	ontrci 4548	. . . . . . . . . 10 ⊢ Tr {𝑥 ∈ {∅} ∣ ¬ 𝜑}
58		treq 4214	. . . . . . . . . 10 ⊢ (𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (Tr 𝑧 ↔ Tr {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
59	57, 58	mpbiri 168	. . . . . . . . 9 ⊢ (𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → Tr 𝑧)
60	55, 59	jaoi 724	. . . . . . . 8 ⊢ ((𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → Tr 𝑧)
61	51, 60	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → Tr 𝑧)
62	61	rgen 2595	. . . . . 6 ⊢ ∀𝑧 ∈ 𝐴 Tr 𝑧
63		dford3 4488	. . . . . 6 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 Tr 𝑧))
64	49, 62, 63	mpbir2an 951	. . . . 5 ⊢ Ord 𝐴
65		prexg 4325	. . . . . . . 8 ⊢ (({𝑥 ∈ {∅} ∣ 𝜑} ∈ V ∧ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ V) → {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ∈ V)
66	40, 22, 65	mp2an 426	. . . . . . 7 ⊢ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ∈ V
67	2, 66	eqeltri 2305	. . . . . 6 ⊢ 𝐴 ∈ V
68	67	elon 4495	. . . . 5 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)
69	64, 68	mpbir 146	. . . 4 ⊢ 𝐴 ∈ On
70		2onn 6754	. . . . . 6 ⊢ 2o ∈ ω
71		nnfi 7127	. . . . . 6 ⊢ (2o ∈ ω → 2o ∈ Fin)
72	70, 71	ax-mp 5	. . . . 5 ⊢ 2o ∈ Fin
73		pm5.19 714	. . . . . . . . . 10 ⊢ ¬ (𝜑 ↔ ¬ 𝜑)
74	13	snm 3812	. . . . . . . . . . 11 ⊢ ∃𝑦 𝑦 ∈ {∅}
75		r19.3rmv 3600	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑦 ∈ {∅} → ((𝜑 ↔ ¬ 𝜑) ↔ ∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑)))
76	74, 75	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝜑 ↔ ¬ 𝜑) ↔ ∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑))
77	73, 76	mtbi 677	. . . . . . . . 9 ⊢ ¬ ∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑)
78		rabbi 2722	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑) ↔ {𝑥 ∈ {∅} ∣ 𝜑} = {𝑥 ∈ {∅} ∣ ¬ 𝜑})
79	77, 78	mtbi 677	. . . . . . . 8 ⊢ ¬ {𝑥 ∈ {∅} ∣ 𝜑} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
80	79	neir 2415	. . . . . . 7 ⊢ {𝑥 ∈ {∅} ∣ 𝜑} ≠ {𝑥 ∈ {∅} ∣ ¬ 𝜑}
81		pr2ne 7489	. . . . . . . 8 ⊢ (({𝑥 ∈ {∅} ∣ 𝜑} ∈ V ∧ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ V) → ({{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ≈ 2o ↔ {𝑥 ∈ {∅} ∣ 𝜑} ≠ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
82	40, 22, 81	mp2an 426	. . . . . . 7 ⊢ ({{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ≈ 2o ↔ {𝑥 ∈ {∅} ∣ 𝜑} ≠ {𝑥 ∈ {∅} ∣ ¬ 𝜑})
83	80, 82	mpbir 146	. . . . . 6 ⊢ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ≈ 2o
84	2, 83	eqbrtri 4130	. . . . 5 ⊢ 𝐴 ≈ 2o
85		enfii 7129	. . . . 5 ⊢ ((2o ∈ Fin ∧ 𝐴 ≈ 2o) → 𝐴 ∈ Fin)
86	72, 84, 85	mp2an 426	. . . 4 ⊢ 𝐴 ∈ Fin
87	69, 86	elini 3403	. . 3 ⊢ 𝐴 ∈ (On ∩ Fin)
88		eleq2 2296	. . 3 ⊢ (ω = (On ∩ Fin) → (𝐴 ∈ ω ↔ 𝐴 ∈ (On ∩ Fin)))
89	87, 88	mpbiri 168	. 2 ⊢ (ω = (On ∩ Fin) → 𝐴 ∈ ω)
90		df1o2 6661	. . . . 5 ⊢ 1o = {∅}
91		1lt2o 6675	. . . . 5 ⊢ 1o ∈ 2o
92	90, 91	eqeltrri 2306	. . . 4 ⊢ {∅} ∈ 2o
93		nneneq 7111	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 2o ∈ ω) → (𝐴 ≈ 2o ↔ 𝐴 = 2o))
94	70, 93	mpan2 425	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ≈ 2o ↔ 𝐴 = 2o))
95	84, 94	mpbii 148	. . . 4 ⊢ (𝐴 ∈ ω → 𝐴 = 2o)
96	92, 95	eleqtrrid 2322	. . 3 ⊢ (𝐴 ∈ ω → {∅} ∈ 𝐴)
97		elpri 3712	. . . 4 ⊢ ({∅} ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} → ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
98	97, 2	eleq2s 2327	. . 3 ⊢ ({∅} ∈ 𝐴 → ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
99	96, 98	syl 14	. 2 ⊢ (𝐴 ∈ ω → ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
100	13	snid 3720	. . . . . . 7 ⊢ ∅ ∈ {∅}
101		eleq2 2296	. . . . . . 7 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → (∅ ∈ {∅} ↔ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
102	100, 101	mpbii 148	. . . . . 6 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
103		biidd 172	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜑))
104	103	elrab 2973	. . . . . 6 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {∅} ∧ 𝜑))
105	102, 104	sylib 122	. . . . 5 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → (∅ ∈ {∅} ∧ 𝜑))
106	105	simprd 114	. . . 4 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → 𝜑)
107		eleq2 2296	. . . . . . 7 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (∅ ∈ {∅} ↔ ∅ ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
108	100, 107	mpbii 148	. . . . . 6 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ∅ ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑})
109		biidd 172	. . . . . . 7 ⊢ (𝑥 = ∅ → (¬ 𝜑 ↔ ¬ 𝜑))
110	109	elrab 2973	. . . . . 6 ⊢ (∅ ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (∅ ∈ {∅} ∧ ¬ 𝜑))
111	108, 110	sylib 122	. . . . 5 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (∅ ∈ {∅} ∧ ¬ 𝜑))
112	111	simprd 114	. . . 4 ⊢ ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
113	106, 112	orim12i 767	. . 3 ⊢ (({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
114		df-dc 843	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
115	113, 114	sylibr 134	. 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 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   = wceq 1398  ∃wex 1541   ∈ wcel 2203   ≠ wne 2412  ∀wral 2520  {crab 2524  Vcvv 2813   ∩ cin 3210  ∅c0 3508  {csn 3689  {cpr 3690   class class class wbr 4109  Tr wtr 4208  Ord word 4483  Oncon0 4484  ωcom 4712  1_oc1o 6640  2_oc2o 6641   ≈ cen 6973  Fincfn 6975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1o 6647  df-2o 6648  df-er 6767  df-en 6976  df-fin 6978

This theorem is referenced by:  exmidonfin  7497