Theorem exmidapne 7258

Description: Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)

Assertion

Ref	Expression
exmidapne	⊢ (EXMID → (𝑅 TAp 𝐴 ↔ 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}))

Distinct variable group:   𝑢,𝐴,𝑣

Allowed substitution hints:   𝑅(𝑣,𝑢)

Proof of Theorem exmidapne

Dummy variables 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 528	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → 𝑅 TAp 𝐴)
2		simpr 110	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → 𝑝 ∈ 𝑅)
3		dftap2 7249	. . . . . . . . . 10 ⊢ (𝑅 TAp 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))))
4	3	biimpi 120	. . . . . . . . 9 ⊢ (𝑅 TAp 𝐴 → (𝑅 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))))
5	4	simp1d 1009	. . . . . . . 8 ⊢ (𝑅 TAp 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))
6	5	sseld 3154	. . . . . . 7 ⊢ (𝑅 TAp 𝐴 → (𝑝 ∈ 𝑅 → 𝑝 ∈ (𝐴 × 𝐴)))
7	1, 2, 6	sylc 62	. . . . . 6 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → 𝑝 ∈ (𝐴 × 𝐴))
8		1st2nd2 6175	. . . . . 6 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
9	7, 8	syl 14	. . . . 5 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
10		xp1st 6165	. . . . . . . 8 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → (1st ‘𝑝) ∈ 𝐴)
11	7, 10	syl 14	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → (1st ‘𝑝) ∈ 𝐴)
12		xp2nd 6166	. . . . . . . 8 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → (2nd ‘𝑝) ∈ 𝐴)
13	7, 12	syl 14	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → (2nd ‘𝑝) ∈ 𝐴)
14	9, 2	eqeltrrd 2255	. . . . . . . . . 10 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ 𝑅)
15	14	adantr 276	. . . . . . . . 9 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ 𝑅)
16		simpr 110	. . . . . . . . . . 11 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → (1st ‘𝑝) = (2nd ‘𝑝))
17	16	opeq2d 3785	. . . . . . . . . 10 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → ⟨(1st ‘𝑝), (1st ‘𝑝)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
18		id 19	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (1st ‘𝑝) → 𝑥 = (1st ‘𝑝))
19	18, 18	breq12d 4016	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥𝑅𝑥 ↔ (1st ‘𝑝)𝑅(1st ‘𝑝)))
20	19	notbid 667	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → (¬ 𝑥𝑅𝑥 ↔ ¬ (1st ‘𝑝)𝑅(1st ‘𝑝)))
21	4	simp2d 1010	. . . . . . . . . . . . . 14 ⊢ (𝑅 TAp 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
22	21	simpld 112	. . . . . . . . . . . . 13 ⊢ (𝑅 TAp 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
23	22	ad3antlr 493	. . . . . . . . . . . 12 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
24	11	adantr 276	. . . . . . . . . . . 12 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → (1st ‘𝑝) ∈ 𝐴)
25	20, 23, 24	rspcdva 2846	. . . . . . . . . . 11 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → ¬ (1st ‘𝑝)𝑅(1st ‘𝑝))
26		df-br 4004	. . . . . . . . . . 11 ⊢ ((1st ‘𝑝)𝑅(1st ‘𝑝) ↔ ⟨(1st ‘𝑝), (1st ‘𝑝)⟩ ∈ 𝑅)
27	25, 26	sylnib 676	. . . . . . . . . 10 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → ¬ ⟨(1st ‘𝑝), (1st ‘𝑝)⟩ ∈ 𝑅)
28	17, 27	eqneltrrd 2274	. . . . . . . . 9 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → ¬ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ 𝑅)
29	15, 28	pm2.65da 661	. . . . . . . 8 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → ¬ (1st ‘𝑝) = (2nd ‘𝑝))
30	29	neqned 2354	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → (1st ‘𝑝) ≠ (2nd ‘𝑝))
31	11, 13, 30	jca31 309	. . . . . 6 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → (((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴) ∧ (1st ‘𝑝) ≠ (2nd ‘𝑝)))
32		eleq1 2240	. . . . . . . . . 10 ⊢ (𝑢 = (1st ‘𝑝) → (𝑢 ∈ 𝐴 ↔ (1st ‘𝑝) ∈ 𝐴))
33		eleq1 2240	. . . . . . . . . 10 ⊢ (𝑣 = (2nd ‘𝑝) → (𝑣 ∈ 𝐴 ↔ (2nd ‘𝑝) ∈ 𝐴))
34	32, 33	bi2anan9 606	. . . . . . . . 9 ⊢ ((𝑢 = (1st ‘𝑝) ∧ 𝑣 = (2nd ‘𝑝)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ↔ ((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴)))
35		simpl 109	. . . . . . . . . 10 ⊢ ((𝑢 = (1st ‘𝑝) ∧ 𝑣 = (2nd ‘𝑝)) → 𝑢 = (1st ‘𝑝))
36		simpr 110	. . . . . . . . . 10 ⊢ ((𝑢 = (1st ‘𝑝) ∧ 𝑣 = (2nd ‘𝑝)) → 𝑣 = (2nd ‘𝑝))
37	35, 36	neeq12d 2367	. . . . . . . . 9 ⊢ ((𝑢 = (1st ‘𝑝) ∧ 𝑣 = (2nd ‘𝑝)) → (𝑢 ≠ 𝑣 ↔ (1st ‘𝑝) ≠ (2nd ‘𝑝)))
38	34, 37	anbi12d 473	. . . . . . . 8 ⊢ ((𝑢 = (1st ‘𝑝) ∧ 𝑣 = (2nd ‘𝑝)) → (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣) ↔ (((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴) ∧ (1st ‘𝑝) ≠ (2nd ‘𝑝))))
39	38	opelopabga 4263	. . . . . . 7 ⊢ (((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴) → (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ (((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴) ∧ (1st ‘𝑝) ≠ (2nd ‘𝑝))))
40	11, 13, 39	syl2anc 411	. . . . . 6 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ↔ (((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴) ∧ (1st ‘𝑝) ≠ (2nd ‘𝑝))))
41	31, 40	mpbird 167	. . . . 5 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
42	9, 41	eqeltrd 2254	. . . 4 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
43		relopab 4753	. . . . . . 7 ⊢ Rel {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}
44		1st2nd 6181	. . . . . . 7 ⊢ ((Rel {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
45	43, 44	mpan 424	. . . . . 6 ⊢ (𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
46	45	adantl 277	. . . . 5 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
47		breq2 4007	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑝) → ((1st ‘𝑝)𝑅𝑦 ↔ (1st ‘𝑝)𝑅(2nd ‘𝑝)))
48	47	notbid 667	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘𝑝) → (¬ (1st ‘𝑝)𝑅𝑦 ↔ ¬ (1st ‘𝑝)𝑅(2nd ‘𝑝)))
49		eqeq2 2187	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘𝑝) → ((1st ‘𝑝) = 𝑦 ↔ (1st ‘𝑝) = (2nd ‘𝑝)))
50	48, 49	imbi12d 234	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘𝑝) → ((¬ (1st ‘𝑝)𝑅𝑦 → (1st ‘𝑝) = 𝑦) ↔ (¬ (1st ‘𝑝)𝑅(2nd ‘𝑝) → (1st ‘𝑝) = (2nd ‘𝑝))))
51		breq1 4006	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥𝑅𝑦 ↔ (1st ‘𝑝)𝑅𝑦))
52	51	notbid 667	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → (¬ 𝑥𝑅𝑦 ↔ ¬ (1st ‘𝑝)𝑅𝑦))
53		eqeq1 2184	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥 = 𝑦 ↔ (1st ‘𝑝) = 𝑦))
54	52, 53	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑝) → ((¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦) ↔ (¬ (1st ‘𝑝)𝑅𝑦 → (1st ‘𝑝) = 𝑦)))
55	54	ralbidv 2477	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑝) → (∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (¬ (1st ‘𝑝)𝑅𝑦 → (1st ‘𝑝) = 𝑦)))
56	4	simp3d 1011	. . . . . . . . . . 11 ⊢ (𝑅 TAp 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦)))
57	56	simprd 114	. . . . . . . . . 10 ⊢ (𝑅 TAp 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))
58	57	ad2antlr 489	. . . . . . . . 9 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))
59	32	anbi1d 465	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑝) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ↔ ((1st ‘𝑝) ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)))
60		neeq1 2360	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑝) → (𝑢 ≠ 𝑣 ↔ (1st ‘𝑝) ≠ 𝑣))
61	59, 60	anbi12d 473	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1st ‘𝑝) → (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣) ↔ (((1st ‘𝑝) ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (1st ‘𝑝) ≠ 𝑣)))
62	33	anbi2d 464	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (2nd ‘𝑝) → (((1st ‘𝑝) ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ↔ ((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴)))
63		neeq2 2361	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (2nd ‘𝑝) → ((1st ‘𝑝) ≠ 𝑣 ↔ (1st ‘𝑝) ≠ (2nd ‘𝑝)))
64	62, 63	anbi12d 473	. . . . . . . . . . . . 13 ⊢ (𝑣 = (2nd ‘𝑝) → ((((1st ‘𝑝) ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (1st ‘𝑝) ≠ 𝑣) ↔ (((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴) ∧ (1st ‘𝑝) ≠ (2nd ‘𝑝))))
65	61, 64	elopabi 6195	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} → (((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴) ∧ (1st ‘𝑝) ≠ (2nd ‘𝑝)))
66	65	adantl 277	. . . . . . . . . . 11 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → (((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴) ∧ (1st ‘𝑝) ≠ (2nd ‘𝑝)))
67	66	simpld 112	. . . . . . . . . 10 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → ((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴))
68	67	simpld 112	. . . . . . . . 9 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → (1st ‘𝑝) ∈ 𝐴)
69	55, 58, 68	rspcdva 2846	. . . . . . . 8 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → ∀𝑦 ∈ 𝐴 (¬ (1st ‘𝑝)𝑅𝑦 → (1st ‘𝑝) = 𝑦))
70	67	simprd 114	. . . . . . . 8 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → (2nd ‘𝑝) ∈ 𝐴)
71	50, 69, 70	rspcdva 2846	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → (¬ (1st ‘𝑝)𝑅(2nd ‘𝑝) → (1st ‘𝑝) = (2nd ‘𝑝)))
72	66	simprd 114	. . . . . . . 8 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → (1st ‘𝑝) ≠ (2nd ‘𝑝))
73	72	neneqd 2368	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → ¬ (1st ‘𝑝) = (2nd ‘𝑝))
74		exmidexmid 4196	. . . . . . . . 9 ⊢ (EXMID → DECID (1st ‘𝑝)𝑅(2nd ‘𝑝))
75		con1dc 856	. . . . . . . . 9 ⊢ (DECID (1st ‘𝑝)𝑅(2nd ‘𝑝) → ((¬ (1st ‘𝑝)𝑅(2nd ‘𝑝) → (1st ‘𝑝) = (2nd ‘𝑝)) → (¬ (1st ‘𝑝) = (2nd ‘𝑝) → (1st ‘𝑝)𝑅(2nd ‘𝑝))))
76	74, 75	syl 14	. . . . . . . 8 ⊢ (EXMID → ((¬ (1st ‘𝑝)𝑅(2nd ‘𝑝) → (1st ‘𝑝) = (2nd ‘𝑝)) → (¬ (1st ‘𝑝) = (2nd ‘𝑝) → (1st ‘𝑝)𝑅(2nd ‘𝑝))))
77	76	ad2antrr 488	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → ((¬ (1st ‘𝑝)𝑅(2nd ‘𝑝) → (1st ‘𝑝) = (2nd ‘𝑝)) → (¬ (1st ‘𝑝) = (2nd ‘𝑝) → (1st ‘𝑝)𝑅(2nd ‘𝑝))))
78	71, 73, 77	mp2d 47	. . . . . 6 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → (1st ‘𝑝)𝑅(2nd ‘𝑝))
79		df-br 4004	. . . . . 6 ⊢ ((1st ‘𝑝)𝑅(2nd ‘𝑝) ↔ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ 𝑅)
80	78, 79	sylib 122	. . . . 5 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ 𝑅)
81	46, 80	eqeltrd 2254	. . . 4 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → 𝑝 ∈ 𝑅)
82	42, 81	impbida 596	. . 3 ⊢ ((EXMID ∧ 𝑅 TAp 𝐴) → (𝑝 ∈ 𝑅 ↔ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}))
83	82	eqrdv 2175	. 2 ⊢ ((EXMID ∧ 𝑅 TAp 𝐴) → 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
84		exmidexmid 4196	. . . . . . 7 ⊢ (EXMID → DECID 𝑥 = 𝑦)
85	84	ralrimivw 2551	. . . . . 6 ⊢ (EXMID → ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
86	85	ralrimivw 2551	. . . . 5 ⊢ (EXMID → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
87		netap 7252	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴)
88	86, 87	syl 14	. . . 4 ⊢ (EXMID → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴)
89	88	adantr 276	. . 3 ⊢ ((EXMID ∧ 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴)
90		tapeq1 7250	. . . 4 ⊢ (𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} → (𝑅 TAp 𝐴 ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴))
91	90	adantl 277	. . 3 ⊢ ((EXMID ∧ 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → (𝑅 TAp 𝐴 ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴))
92	89, 91	mpbird 167	. 2 ⊢ ((EXMID ∧ 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → 𝑅 TAp 𝐴)
93	83, 92	impbida 596	1 ⊢ (EXMID → (𝑅 TAp 𝐴 ↔ 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455   ⊆ wss 3129  ⟨cop 3595   class class class wbr 4003  {copab 4063  EXMIDwem 4194   × cxp 4624  Rel wrel 4631  ‘cfv 5216  1^st c1st 6138  2^nd c2nd 6139   TAp wtap 7247

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-exmid 4195  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224  df-1st 6140  df-2nd 6141  df-pap 7246  df-tap 7248

This theorem is referenced by:  exmidmotap  7259