Theorem exmidapne 7616

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 533	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → 𝑅 TAp 𝐴)
2		simpr 110	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → 𝑝 ∈ 𝑅)
3		dftap2 7607	. . . . . . . . . 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 1040	. . . . . . . 8 ⊢ (𝑅 TAp 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))
6	5	sseld 3247	. . . . . . 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 6399	. . . . . 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 6389	. . . . . . . 8 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → (1st ‘𝑝) ∈ 𝐴)
11	7, 10	syl 14	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → (1st ‘𝑝) ∈ 𝐴)
12		xp2nd 6390	. . . . . . . 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 2316	. . . . . . . . . 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 3906	. . . . . . . . . 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 4138	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥𝑅𝑥 ↔ (1st ‘𝑝)𝑅(1st ‘𝑝)))
20	19	notbid 677	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → (¬ 𝑥𝑅𝑥 ↔ ¬ (1st ‘𝑝)𝑅(1st ‘𝑝)))
21	4	simp2d 1041	. . . . . . . . . . . . . 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 497	. . . . . . . . . . . 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 2934	. . . . . . . . . . 11 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → ¬ (1st ‘𝑝)𝑅(1st ‘𝑝))
26		df-br 4126	. . . . . . . . . . 11 ⊢ ((1st ‘𝑝)𝑅(1st ‘𝑝) ↔ ⟨(1st ‘𝑝), (1st ‘𝑝)⟩ ∈ 𝑅)
27	25, 26	sylnib 687	. . . . . . . . . 10 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → ¬ ⟨(1st ‘𝑝), (1st ‘𝑝)⟩ ∈ 𝑅)
28	17, 27	eqneltrrd 2335	. . . . . . . . 9 ⊢ ((((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) ∧ (1st ‘𝑝) = (2nd ‘𝑝)) → ¬ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ 𝑅)
29	15, 28	pm2.65da 671	. . . . . . . 8 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → ¬ (1st ‘𝑝) = (2nd ‘𝑝))
30	29	neqned 2427	. . . . . . 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 2301	. . . . . . . . . 10 ⊢ (𝑢 = (1st ‘𝑝) → (𝑢 ∈ 𝐴 ↔ (1st ‘𝑝) ∈ 𝐴))
33		eleq1 2301	. . . . . . . . . 10 ⊢ (𝑣 = (2nd ‘𝑝) → (𝑣 ∈ 𝐴 ↔ (2nd ‘𝑝) ∈ 𝐴))
34	32, 33	bi2anan9 614	. . . . . . . . 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 2440	. . . . . . . . 9 ⊢ ((𝑢 = (1st ‘𝑝) ∧ 𝑣 = (2nd ‘𝑝)) → (𝑢 ≠ 𝑣 ↔ (1st ‘𝑝) ≠ (2nd ‘𝑝)))
38	34, 37	anbi12d 477	. . . . . . . 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 4400	. . . . . . 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 415	. . . . . 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 2315	. . . 4 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ 𝑅) → 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
43		relopab 4901	. . . . . . 7 ⊢ Rel {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}
44		1st2nd 6405	. . . . . . 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 428	. . . . . 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 4129	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑝) → ((1st ‘𝑝)𝑅𝑦 ↔ (1st ‘𝑝)𝑅(2nd ‘𝑝)))
48	47	notbid 677	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘𝑝) → (¬ (1st ‘𝑝)𝑅𝑦 ↔ ¬ (1st ‘𝑝)𝑅(2nd ‘𝑝)))
49		eqeq2 2248	. . . . . . . . 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 4128	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑝) → (𝑥𝑅𝑦 ↔ (1st ‘𝑝)𝑅𝑦))
52	51	notbid 677	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑝) → (¬ 𝑥𝑅𝑦 ↔ ¬ (1st ‘𝑝)𝑅𝑦))
53		eqeq1 2245	. . . . . . . . . . 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 2550	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑝) → (∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (¬ (1st ‘𝑝)𝑅𝑦 → (1st ‘𝑝) = 𝑦)))
56	4	simp3d 1042	. . . . . . . . . . 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 493	. . . . . . . . 9 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))
59	32	anbi1d 469	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑝) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ↔ ((1st ‘𝑝) ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)))
60		neeq1 2433	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (1st ‘𝑝) → (𝑢 ≠ 𝑣 ↔ (1st ‘𝑝) ≠ 𝑣))
61	59, 60	anbi12d 477	. . . . . . . . . . . . 13 ⊢ (𝑢 = (1st ‘𝑝) → (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣) ↔ (((1st ‘𝑝) ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (1st ‘𝑝) ≠ 𝑣)))
62	33	anbi2d 468	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (2nd ‘𝑝) → (((1st ‘𝑝) ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ↔ ((1st ‘𝑝) ∈ 𝐴 ∧ (2nd ‘𝑝) ∈ 𝐴)))
63		neeq2 2434	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (2nd ‘𝑝) → ((1st ‘𝑝) ≠ 𝑣 ↔ (1st ‘𝑝) ≠ (2nd ‘𝑝)))
64	62, 63	anbi12d 477	. . . . . . . . . . . . 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 6421	. . . . . . . . . . . 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 2934	. . . . . . . 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 2934	. . . . . . 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 2441	. . . . . . 7 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → ¬ (1st ‘𝑝) = (2nd ‘𝑝))
74		exmidexmid 4328	. . . . . . . . 9 ⊢ (EXMID → DECID (1st ‘𝑝)𝑅(2nd ‘𝑝))
75		con1dc 868	. . . . . . . . 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 492	. . . . . . 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 4126	. . . . . 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 2315	. . . 4 ⊢ (((EXMID ∧ 𝑅 TAp 𝐴) ∧ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}) → 𝑝 ∈ 𝑅)
82	42, 81	impbida 604	. . 3 ⊢ ((EXMID ∧ 𝑅 TAp 𝐴) → (𝑝 ∈ 𝑅 ↔ 𝑝 ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}))
83	82	eqrdv 2236	. 2 ⊢ ((EXMID ∧ 𝑅 TAp 𝐴) → 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)})
84		exmidexmid 4328	. . . . . . 7 ⊢ (EXMID → DECID 𝑥 = 𝑦)
85	84	ralrimivw 2624	. . . . . 6 ⊢ (EXMID → ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
86	85	ralrimivw 2624	. . . . 5 ⊢ (EXMID → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
87		netap 7610	. . . . 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 7608	. . . 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 604	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 720  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528   ⊆ wss 3220  ⟨cop 3708   class class class wbr 4125  {copab 4186  EXMIDwem 4326   × cxp 4767  Rel wrel 4774  ‘cfv 5372  1^st c1st 6362  2^nd c2nd 6363   TAp wtap 7604

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-exmid 4327  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-1st 6364  df-2nd 6365  df-pap 7598  df-tap 7605

This theorem is referenced by:  exmidmotap  7617