Theorem exmidaclem 7164

Description: Lemma for exmidac 7165. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)

Hypotheses

Ref	Expression
exmidaclem.a	⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}
exmidaclem.b	⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}
exmidaclem.c	⊢ 𝐶 = {𝐴, 𝐵}

Assertion

Ref	Expression
exmidaclem	⊢ (CHOICE → EXMID)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem exmidaclem

Dummy variables 𝑧 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 108	. . . 4 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → CHOICE)
2		exmidaclem.c	. . . . . 6 ⊢ 𝐶 = {𝐴, 𝐵}
3		exmidaclem.a	. . . . . . . 8 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}
4		pp0ex 4168	. . . . . . . . 9 ⊢ {∅, {∅}} ∈ V
5	4	rabex 4126	. . . . . . . 8 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})} ∈ V
6	3, 5	eqeltri 2239	. . . . . . 7 ⊢ 𝐴 ∈ V
7		exmidaclem.b	. . . . . . . 8 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}
8	4	rabex 4126	. . . . . . . 8 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})} ∈ V
9	7, 8	eqeltri 2239	. . . . . . 7 ⊢ 𝐵 ∈ V
10		prexg 4189	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
11	6, 9, 10	mp2an 423	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
12	2, 11	eqeltri 2239	. . . . 5 ⊢ 𝐶 ∈ V
13	12	a1i 9	. . . 4 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → 𝐶 ∈ V)
14		simpr 109	. . . . . . 7 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐶)
15	14, 2	eleqtrdi 2259	. . . . . 6 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ {𝐴, 𝐵})
16		elpri 3599	. . . . . 6 ⊢ (𝑧 ∈ {𝐴, 𝐵} → (𝑧 = 𝐴 ∨ 𝑧 = 𝐵))
17		0ex 4109	. . . . . . . . . . 11 ⊢ ∅ ∈ V
18	17	prid1 3682	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, {∅}}
19		eqid 2165	. . . . . . . . . . 11 ⊢ ∅ = ∅
20	19	orci 721	. . . . . . . . . 10 ⊢ (∅ = ∅ ∨ 𝑦 = {∅})
21		eqeq1 2172	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
22	21	orbi1d 781	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑦 = {∅}) ↔ (∅ = ∅ ∨ 𝑦 = {∅})))
23	22, 3	elrab2 2885	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = ∅ ∨ 𝑦 = {∅})))
24	18, 20, 23	mpbir2an 932	. . . . . . . . 9 ⊢ ∅ ∈ 𝐴
25		eleq2 2230	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝐴))
26	24, 25	mpbiri 167	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ∅ ∈ 𝑧)
27		elex2 2742	. . . . . . . 8 ⊢ (∅ ∈ 𝑧 → ∃𝑤 𝑤 ∈ 𝑧)
28	26, 27	syl 14	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ∃𝑤 𝑤 ∈ 𝑧)
29		p0ex 4167	. . . . . . . . . . 11 ⊢ {∅} ∈ V
30	29	prid2 3683	. . . . . . . . . 10 ⊢ {∅} ∈ {∅, {∅}}
31		eqid 2165	. . . . . . . . . . 11 ⊢ {∅} = {∅}
32	31	orci 721	. . . . . . . . . 10 ⊢ ({∅} = {∅} ∨ 𝑦 = {∅})
33		eqeq1 2172	. . . . . . . . . . . 12 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
34	33	orbi1d 781	. . . . . . . . . . 11 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ 𝑦 = {∅}) ↔ ({∅} = {∅} ∨ 𝑦 = {∅})))
35	34, 7	elrab2 2885	. . . . . . . . . 10 ⊢ ({∅} ∈ 𝐵 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ 𝑦 = {∅})))
36	30, 32, 35	mpbir2an 932	. . . . . . . . 9 ⊢ {∅} ∈ 𝐵
37		eleq2 2230	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → ({∅} ∈ 𝑧 ↔ {∅} ∈ 𝐵))
38	36, 37	mpbiri 167	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → {∅} ∈ 𝑧)
39		elex2 2742	. . . . . . . 8 ⊢ ({∅} ∈ 𝑧 → ∃𝑤 𝑤 ∈ 𝑧)
40	38, 39	syl 14	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ∃𝑤 𝑤 ∈ 𝑧)
41	28, 40	jaoi 706	. . . . . 6 ⊢ ((𝑧 = 𝐴 ∨ 𝑧 = 𝐵) → ∃𝑤 𝑤 ∈ 𝑧)
42	15, 16, 41	3syl 17	. . . . 5 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ 𝑧 ∈ 𝐶) → ∃𝑤 𝑤 ∈ 𝑧)
43	42	ralrimiva 2539	. . . 4 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → ∀𝑧 ∈ 𝐶 ∃𝑤 𝑤 ∈ 𝑧)
44	1, 13, 43	acfun 7163	. . 3 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → ∃𝑓(𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧))
45		0nep0 4144	. . . . . . . . . 10 ⊢ ∅ ≠ {∅}
46	45	neii 2338	. . . . . . . . 9 ⊢ ¬ ∅ = {∅}
47		simplr 520	. . . . . . . . . 10 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → (𝑓‘𝐴) = ∅)
48		simpr 109	. . . . . . . . . 10 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → (𝑓‘𝐵) = {∅})
49	47, 48	eqeq12d 2180	. . . . . . . . 9 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → ((𝑓‘𝐴) = (𝑓‘𝐵) ↔ ∅ = {∅}))
50	46, 49	mtbiri 665	. . . . . . . 8 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → ¬ (𝑓‘𝐴) = (𝑓‘𝐵))
51		olc 701	. . . . . . . . . . . . 13 ⊢ (𝑦 = {∅} → (𝑥 = ∅ ∨ 𝑦 = {∅}))
52	51	ralrimivw 2540	. . . . . . . . . . . 12 ⊢ (𝑦 = {∅} → ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝑦 = {∅}))
53		rabid2 2642	. . . . . . . . . . . 12 ⊢ ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝑦 = {∅}))
54	52, 53	sylibr 133	. . . . . . . . . . 11 ⊢ (𝑦 = {∅} → {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})})
55	54, 3	eqtr4di 2217	. . . . . . . . . 10 ⊢ (𝑦 = {∅} → {∅, {∅}} = 𝐴)
56		olc 701	. . . . . . . . . . . . 13 ⊢ (𝑦 = {∅} → (𝑥 = {∅} ∨ 𝑦 = {∅}))
57	56	ralrimivw 2540	. . . . . . . . . . . 12 ⊢ (𝑦 = {∅} → ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝑦 = {∅}))
58		rabid2 2642	. . . . . . . . . . . 12 ⊢ ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝑦 = {∅}))
59	57, 58	sylibr 133	. . . . . . . . . . 11 ⊢ (𝑦 = {∅} → {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})})
60	59, 7	eqtr4di 2217	. . . . . . . . . 10 ⊢ (𝑦 = {∅} → {∅, {∅}} = 𝐵)
61	55, 60	eqtr3d 2200	. . . . . . . . 9 ⊢ (𝑦 = {∅} → 𝐴 = 𝐵)
62	61	fveq2d 5490	. . . . . . . 8 ⊢ (𝑦 = {∅} → (𝑓‘𝐴) = (𝑓‘𝐵))
63	50, 62	nsyl 618	. . . . . . 7 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → ¬ 𝑦 = {∅})
64	63	olcd 724	. . . . . 6 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
65		simpr 109	. . . . . . 7 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ 𝑦 = {∅}) → 𝑦 = {∅})
66	65	orcd 723	. . . . . 6 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ 𝑦 = {∅}) → (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
67		fveq2 5486	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → (𝑓‘𝑧) = (𝑓‘𝐵))
68		id 19	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
69	67, 68	eleq12d 2237	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝐵) ∈ 𝐵))
70		simprr 522	. . . . . . . . . 10 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)
71	9	prid2 3683	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ {𝐴, 𝐵}
72	71, 2	eleqtrri 2242	. . . . . . . . . . 11 ⊢ 𝐵 ∈ 𝐶
73	72	a1i 9	. . . . . . . . . 10 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → 𝐵 ∈ 𝐶)
74	69, 70, 73	rspcdva 2835	. . . . . . . . 9 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → (𝑓‘𝐵) ∈ 𝐵)
75		eqeq1 2172	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝐵) → (𝑥 = {∅} ↔ (𝑓‘𝐵) = {∅}))
76	75	orbi1d 781	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝐵) → ((𝑥 = {∅} ∨ 𝑦 = {∅}) ↔ ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅})))
77	76, 7	elrab2 2885	. . . . . . . . 9 ⊢ ((𝑓‘𝐵) ∈ 𝐵 ↔ ((𝑓‘𝐵) ∈ {∅, {∅}} ∧ ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅})))
78	74, 77	sylib 121	. . . . . . . 8 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘𝐵) ∈ {∅, {∅}} ∧ ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅})))
79	78	simprd 113	. . . . . . 7 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅}))
80	79	adantr 274	. . . . . 6 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) → ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅}))
81	64, 66, 80	mpjaodan 788	. . . . 5 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) → (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
82		df-dc 825	. . . . 5 ⊢ (DECID 𝑦 = {∅} ↔ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
83	81, 82	sylibr 133	. . . 4 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) → DECID 𝑦 = {∅})
84		simpr 109	. . . . . 6 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ 𝑦 = {∅}) → 𝑦 = {∅})
85	84	orcd 723	. . . . 5 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ 𝑦 = {∅}) → (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
86	85, 82	sylibr 133	. . . 4 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ 𝑦 = {∅}) → DECID 𝑦 = {∅})
87		fveq2 5486	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑓‘𝑧) = (𝑓‘𝐴))
88		id 19	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → 𝑧 = 𝐴)
89	87, 88	eleq12d 2237	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝐴) ∈ 𝐴))
90	6	prid1 3682	. . . . . . . . 9 ⊢ 𝐴 ∈ {𝐴, 𝐵}
91	90, 2	eleqtrri 2242	. . . . . . . 8 ⊢ 𝐴 ∈ 𝐶
92	91	a1i 9	. . . . . . 7 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → 𝐴 ∈ 𝐶)
93	89, 70, 92	rspcdva 2835	. . . . . 6 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → (𝑓‘𝐴) ∈ 𝐴)
94		eqeq1 2172	. . . . . . . 8 ⊢ (𝑥 = (𝑓‘𝐴) → (𝑥 = ∅ ↔ (𝑓‘𝐴) = ∅))
95	94	orbi1d 781	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝐴) → ((𝑥 = ∅ ∨ 𝑦 = {∅}) ↔ ((𝑓‘𝐴) = ∅ ∨ 𝑦 = {∅})))
96	95, 3	elrab2 2885	. . . . . 6 ⊢ ((𝑓‘𝐴) ∈ 𝐴 ↔ ((𝑓‘𝐴) ∈ {∅, {∅}} ∧ ((𝑓‘𝐴) = ∅ ∨ 𝑦 = {∅})))
97	93, 96	sylib 121	. . . . 5 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘𝐴) ∈ {∅, {∅}} ∧ ((𝑓‘𝐴) = ∅ ∨ 𝑦 = {∅})))
98	97	simprd 113	. . . 4 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘𝐴) = ∅ ∨ 𝑦 = {∅}))
99	83, 86, 98	mpjaodan 788	. . 3 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → DECID 𝑦 = {∅})
100	44, 99	exlimddv 1886	. 2 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → DECID 𝑦 = {∅})
101	100	exmid1dc 4179	1 ⊢ (CHOICE → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 824   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  {crab 2448  Vcvv 2726   ⊆ wss 3116  ∅c0 3409  {csn 3576  {cpr 3577  EXMIDwem 4173   Fn wfn 5183  ‘cfv 5188  CHOICEwac 7161

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-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-dc 825  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-csb 3046  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-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-exmid 4174  df-id 4271  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-ac 7162

This theorem is referenced by:  exmidac  7165