Theorem exmidaclem 7268

Description: Lemma for exmidac 7269. 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 109	. . . 4 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → CHOICE)
2		exmidaclem.c	. . . . . 6 ⊢ 𝐶 = {𝐴, 𝐵}
3		exmidaclem.a	. . . . . . . 8 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}
4		pp0ex 4218	. . . . . . . . 9 ⊢ {∅, {∅}} ∈ V
5	4	rabex 4173	. . . . . . . 8 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})} ∈ V
6	3, 5	eqeltri 2266	. . . . . . 7 ⊢ 𝐴 ∈ V
7		exmidaclem.b	. . . . . . . 8 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}
8	4	rabex 4173	. . . . . . . 8 ⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})} ∈ V
9	7, 8	eqeltri 2266	. . . . . . 7 ⊢ 𝐵 ∈ V
10		prexg 4240	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
11	6, 9, 10	mp2an 426	. . . . . 6 ⊢ {𝐴, 𝐵} ∈ V
12	2, 11	eqeltri 2266	. . . . 5 ⊢ 𝐶 ∈ V
13	12	a1i 9	. . . 4 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → 𝐶 ∈ V)
14		simpr 110	. . . . . . 7 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐶)
15	14, 2	eleqtrdi 2286	. . . . . 6 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ {𝐴, 𝐵})
16		elpri 3641	. . . . . 6 ⊢ (𝑧 ∈ {𝐴, 𝐵} → (𝑧 = 𝐴 ∨ 𝑧 = 𝐵))
17		0ex 4156	. . . . . . . . . . 11 ⊢ ∅ ∈ V
18	17	prid1 3724	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, {∅}}
19		eqid 2193	. . . . . . . . . . 11 ⊢ ∅ = ∅
20	19	orci 732	. . . . . . . . . 10 ⊢ (∅ = ∅ ∨ 𝑦 = {∅})
21		eqeq1 2200	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
22	21	orbi1d 792	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝑦 = {∅}) ↔ (∅ = ∅ ∨ 𝑦 = {∅})))
23	22, 3	elrab2 2919	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = ∅ ∨ 𝑦 = {∅})))
24	18, 20, 23	mpbir2an 944	. . . . . . . . 9 ⊢ ∅ ∈ 𝐴
25		eleq2 2257	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝐴))
26	24, 25	mpbiri 168	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ∅ ∈ 𝑧)
27		elex2 2776	. . . . . . . 8 ⊢ (∅ ∈ 𝑧 → ∃𝑤 𝑤 ∈ 𝑧)
28	26, 27	syl 14	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ∃𝑤 𝑤 ∈ 𝑧)
29		p0ex 4217	. . . . . . . . . . 11 ⊢ {∅} ∈ V
30	29	prid2 3725	. . . . . . . . . 10 ⊢ {∅} ∈ {∅, {∅}}
31		eqid 2193	. . . . . . . . . . 11 ⊢ {∅} = {∅}
32	31	orci 732	. . . . . . . . . 10 ⊢ ({∅} = {∅} ∨ 𝑦 = {∅})
33		eqeq1 2200	. . . . . . . . . . . 12 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
34	33	orbi1d 792	. . . . . . . . . . 11 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ 𝑦 = {∅}) ↔ ({∅} = {∅} ∨ 𝑦 = {∅})))
35	34, 7	elrab2 2919	. . . . . . . . . 10 ⊢ ({∅} ∈ 𝐵 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ 𝑦 = {∅})))
36	30, 32, 35	mpbir2an 944	. . . . . . . . 9 ⊢ {∅} ∈ 𝐵
37		eleq2 2257	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → ({∅} ∈ 𝑧 ↔ {∅} ∈ 𝐵))
38	36, 37	mpbiri 168	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → {∅} ∈ 𝑧)
39		elex2 2776	. . . . . . . 8 ⊢ ({∅} ∈ 𝑧 → ∃𝑤 𝑤 ∈ 𝑧)
40	38, 39	syl 14	. . . . . . 7 ⊢ (𝑧 = 𝐵 → ∃𝑤 𝑤 ∈ 𝑧)
41	28, 40	jaoi 717	. . . . . 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 2567	. . . 4 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → ∀𝑧 ∈ 𝐶 ∃𝑤 𝑤 ∈ 𝑧)
44	1, 13, 43	acfun 7267	. . 3 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → ∃𝑓(𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧))
45		0nep0 4194	. . . . . . . . . 10 ⊢ ∅ ≠ {∅}
46	45	neii 2366	. . . . . . . . 9 ⊢ ¬ ∅ = {∅}
47		simplr 528	. . . . . . . . . 10 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → (𝑓‘𝐴) = ∅)
48		simpr 110	. . . . . . . . . 10 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → (𝑓‘𝐵) = {∅})
49	47, 48	eqeq12d 2208	. . . . . . . . 9 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → ((𝑓‘𝐴) = (𝑓‘𝐵) ↔ ∅ = {∅}))
50	46, 49	mtbiri 676	. . . . . . . 8 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → ¬ (𝑓‘𝐴) = (𝑓‘𝐵))
51		olc 712	. . . . . . . . . . . . 13 ⊢ (𝑦 = {∅} → (𝑥 = ∅ ∨ 𝑦 = {∅}))
52	51	ralrimivw 2568	. . . . . . . . . . . 12 ⊢ (𝑦 = {∅} → ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝑦 = {∅}))
53		rabid2 2671	. . . . . . . . . . . 12 ⊢ ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝑦 = {∅}))
54	52, 53	sylibr 134	. . . . . . . . . . 11 ⊢ (𝑦 = {∅} → {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})})
55	54, 3	eqtr4di 2244	. . . . . . . . . 10 ⊢ (𝑦 = {∅} → {∅, {∅}} = 𝐴)
56		olc 712	. . . . . . . . . . . . 13 ⊢ (𝑦 = {∅} → (𝑥 = {∅} ∨ 𝑦 = {∅}))
57	56	ralrimivw 2568	. . . . . . . . . . . 12 ⊢ (𝑦 = {∅} → ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝑦 = {∅}))
58		rabid2 2671	. . . . . . . . . . . 12 ⊢ ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝑦 = {∅}))
59	57, 58	sylibr 134	. . . . . . . . . . 11 ⊢ (𝑦 = {∅} → {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})})
60	59, 7	eqtr4di 2244	. . . . . . . . . 10 ⊢ (𝑦 = {∅} → {∅, {∅}} = 𝐵)
61	55, 60	eqtr3d 2228	. . . . . . . . 9 ⊢ (𝑦 = {∅} → 𝐴 = 𝐵)
62	61	fveq2d 5558	. . . . . . . 8 ⊢ (𝑦 = {∅} → (𝑓‘𝐴) = (𝑓‘𝐵))
63	50, 62	nsyl 629	. . . . . . 7 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → ¬ 𝑦 = {∅})
64	63	olcd 735	. . . . . 6 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ (𝑓‘𝐵) = {∅}) → (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
65		simpr 110	. . . . . . 7 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ 𝑦 = {∅}) → 𝑦 = {∅})
66	65	orcd 734	. . . . . 6 ⊢ (((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) ∧ 𝑦 = {∅}) → (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
67		fveq2 5554	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → (𝑓‘𝑧) = (𝑓‘𝐵))
68		id 19	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
69	67, 68	eleq12d 2264	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝐵) ∈ 𝐵))
70		simprr 531	. . . . . . . . . 10 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)
71	9	prid2 3725	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ {𝐴, 𝐵}
72	71, 2	eleqtrri 2269	. . . . . . . . . . 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 2869	. . . . . . . . 9 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → (𝑓‘𝐵) ∈ 𝐵)
75		eqeq1 2200	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝐵) → (𝑥 = {∅} ↔ (𝑓‘𝐵) = {∅}))
76	75	orbi1d 792	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝐵) → ((𝑥 = {∅} ∨ 𝑦 = {∅}) ↔ ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅})))
77	76, 7	elrab2 2919	. . . . . . . . 9 ⊢ ((𝑓‘𝐵) ∈ 𝐵 ↔ ((𝑓‘𝐵) ∈ {∅, {∅}} ∧ ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅})))
78	74, 77	sylib 122	. . . . . . . 8 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘𝐵) ∈ {∅, {∅}} ∧ ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅})))
79	78	simprd 114	. . . . . . 7 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅}))
80	79	adantr 276	. . . . . 6 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) → ((𝑓‘𝐵) = {∅} ∨ 𝑦 = {∅}))
81	64, 66, 80	mpjaodan 799	. . . . 5 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) → (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
82		df-dc 836	. . . . 5 ⊢ (DECID 𝑦 = {∅} ↔ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
83	81, 82	sylibr 134	. . . 4 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑓‘𝐴) = ∅) → DECID 𝑦 = {∅})
84		simpr 110	. . . . . 6 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ 𝑦 = {∅}) → 𝑦 = {∅})
85	84	orcd 734	. . . . 5 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ 𝑦 = {∅}) → (𝑦 = {∅} ∨ ¬ 𝑦 = {∅}))
86	85, 82	sylibr 134	. . . 4 ⊢ ((((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) ∧ 𝑦 = {∅}) → DECID 𝑦 = {∅})
87		fveq2 5554	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑓‘𝑧) = (𝑓‘𝐴))
88		id 19	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → 𝑧 = 𝐴)
89	87, 88	eleq12d 2264	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝐴) ∈ 𝐴))
90	6	prid1 3724	. . . . . . . . 9 ⊢ 𝐴 ∈ {𝐴, 𝐵}
91	90, 2	eleqtrri 2269	. . . . . . . 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 2869	. . . . . 6 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → (𝑓‘𝐴) ∈ 𝐴)
94		eqeq1 2200	. . . . . . . 8 ⊢ (𝑥 = (𝑓‘𝐴) → (𝑥 = ∅ ↔ (𝑓‘𝐴) = ∅))
95	94	orbi1d 792	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝐴) → ((𝑥 = ∅ ∨ 𝑦 = {∅}) ↔ ((𝑓‘𝐴) = ∅ ∨ 𝑦 = {∅})))
96	95, 3	elrab2 2919	. . . . . 6 ⊢ ((𝑓‘𝐴) ∈ 𝐴 ↔ ((𝑓‘𝐴) ∈ {∅, {∅}} ∧ ((𝑓‘𝐴) = ∅ ∨ 𝑦 = {∅})))
97	93, 96	sylib 122	. . . . 5 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘𝐴) ∈ {∅, {∅}} ∧ ((𝑓‘𝐴) = ∅ ∨ 𝑦 = {∅})))
98	97	simprd 114	. . . 4 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘𝐴) = ∅ ∨ 𝑦 = {∅}))
99	83, 86, 98	mpjaodan 799	. . 3 ⊢ (((CHOICE ∧ 𝑦 ⊆ {∅}) ∧ (𝑓 Fn 𝐶 ∧ ∀𝑧 ∈ 𝐶 (𝑓‘𝑧) ∈ 𝑧)) → DECID 𝑦 = {∅})
100	44, 99	exlimddv 1910	. 2 ⊢ ((CHOICE ∧ 𝑦 ⊆ {∅}) → DECID 𝑦 = {∅})
101	100	exmid1dc 4229	1 ⊢ (CHOICE → EXMID)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  {crab 2476  Vcvv 2760   ⊆ wss 3153  ∅c0 3446  {csn 3618  {cpr 3619  EXMIDwem 4223   Fn wfn 5249  ‘cfv 5254  CHOICEwac 7265

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-exmid 4224  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ac 7266

This theorem is referenced by:  exmidac  7269