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Theorem hashtpg 13121

Description: The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.)

**Assertion**
Ref	Expression
hashtpg	⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (#‘{𝐴, 𝐵, 𝐶}) = 3))

**Proof of Theorem hashtpg**
Step	Hyp	Ref	Expression
1		simpl3 1059	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → 𝐶 ∈ 𝑊)
2		prfi 8120	. . . . . . 7 ⊢ {𝐴, 𝐵} ∈ Fin
3	2	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → {𝐴, 𝐵} ∈ Fin)
4		elprg 4144	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
5		orcom 401	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐴))
6		nne 2786	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
7		eqcom 2617	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
8	6, 7	bitr2i 264	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐵 ↔ ¬ 𝐵 ≠ 𝐶)
9		nne 2786	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐶 ≠ 𝐴 ↔ 𝐶 = 𝐴)
10	9	bicomi 213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐴 ↔ ¬ 𝐶 ≠ 𝐴)
11	8, 10	orbi12i 542	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐵 ∨ 𝐶 = 𝐴) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
12	5, 11	bitri 263	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
13	4, 12	syl6bb 275	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
14	13	biimpd 218	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
15	14	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
16	15	imp 444	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝐶 ∈ {𝐴, 𝐵}) → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
17	16	olcd 407	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝐶 ∈ {𝐴, 𝐵}) → (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
18	17	ex 449	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))))
19		3orass 1034	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴) ↔ (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
20	18, 19	syl6ibr 241	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
21		3ianor 1048	. . . . . . . . 9 ⊢ (¬ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
22	20, 21	syl6ibr 241	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → ¬ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)))
23	22	con2d 128	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → ¬ 𝐶 ∈ {𝐴, 𝐵}))
24	23	imp 444	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ¬ 𝐶 ∈ {𝐴, 𝐵})
25		hashunsng 13042	. . . . . . 7 ⊢ (𝐶 ∈ 𝑊 → (({𝐴, 𝐵} ∈ Fin ∧ ¬ 𝐶 ∈ {𝐴, 𝐵}) → (#‘({𝐴, 𝐵} ∪ {𝐶})) = ((#‘{𝐴, 𝐵}) + 1)))
26	25	imp 444	. . . . . 6 ⊢ ((𝐶 ∈ 𝑊 ∧ ({𝐴, 𝐵} ∈ Fin ∧ ¬ 𝐶 ∈ {𝐴, 𝐵})) → (#‘({𝐴, 𝐵} ∪ {𝐶})) = ((#‘{𝐴, 𝐵}) + 1))
27	1, 3, 24, 26	syl12anc 1316	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (#‘({𝐴, 𝐵} ∪ {𝐶})) = ((#‘{𝐴, 𝐵}) + 1))
28		simpr1 1060	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → 𝐴 ≠ 𝐵)
29		3simpa 1051	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
30	29	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
31		hashprg 13043	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → (𝐴 ≠ 𝐵 ↔ (#‘{𝐴, 𝐵}) = 2))
32	30, 31	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (𝐴 ≠ 𝐵 ↔ (#‘{𝐴, 𝐵}) = 2))
33	28, 32	mpbid 221	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (#‘{𝐴, 𝐵}) = 2)
34	33	oveq1d 6564	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ((#‘{𝐴, 𝐵}) + 1) = (2 + 1))
35	27, 34	eqtrd 2644	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (#‘({𝐴, 𝐵} ∪ {𝐶})) = (2 + 1))
36		df-tp 4130	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
37	36	fveq2i 6106	. . . 4 ⊢ (#‘{𝐴, 𝐵, 𝐶}) = (#‘({𝐴, 𝐵} ∪ {𝐶}))
38		df-3 10957	. . . 4 ⊢ 3 = (2 + 1)
39	35, 37, 38	3eqtr4g 2669	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (#‘{𝐴, 𝐵, 𝐶}) = 3)
40	39	ex 449	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → (#‘{𝐴, 𝐵, 𝐶}) = 3))
41		nne 2786	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
42		hashprlei 13107	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ Fin ∧ (#‘{𝐵, 𝐶}) ≤ 2)
43		prfi 8120	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐶} ∈ Fin
44		hashcl 13009	. . . . . . . . . . . . . . 15 ⊢ ({𝐵, 𝐶} ∈ Fin → (#‘{𝐵, 𝐶}) ∈ ℕ₀)
45	44	nn0zd 11356	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (#‘{𝐵, 𝐶}) ∈ ℤ)
46	43, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐵, 𝐶}) ∈ ℤ
47		2z 11286	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
48		zleltp1 11305	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐵, 𝐶}) ≤ 2 ↔ (#‘{𝐵, 𝐶}) < (2 + 1)))
49		2p1e3 11028	. . . . . . . . . . . . . . . . 17 ⊢ (2 + 1) = 3
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → (2 + 1) = 3)
51	50	breq2d 4595	. . . . . . . . . . . . . . 15 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐵, 𝐶}) < (2 + 1) ↔ (#‘{𝐵, 𝐶}) < 3))
52	51	biimpd 218	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐵, 𝐶}) < (2 + 1) → (#‘{𝐵, 𝐶}) < 3))
53	48, 52	sylbid 229	. . . . . . . . . . . . 13 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐵, 𝐶}) ≤ 2 → (#‘{𝐵, 𝐶}) < 3))
54	46, 47, 53	mp2an 704	. . . . . . . . . . . 12 ⊢ ((#‘{𝐵, 𝐶}) ≤ 2 → (#‘{𝐵, 𝐶}) < 3)
55	44	nn0red 11229	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (#‘{𝐵, 𝐶}) ∈ ℝ)
56	43, 55	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐵, 𝐶}) ∈ ℝ
57		3re 10971	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
58	56, 57	ltnei 10040	. . . . . . . . . . . 12 ⊢ ((#‘{𝐵, 𝐶}) < 3 → 3 ≠ (#‘{𝐵, 𝐶}))
59	54, 58	syl 17	. . . . . . . . . . 11 ⊢ ((#‘{𝐵, 𝐶}) ≤ 2 → 3 ≠ (#‘{𝐵, 𝐶}))
60	59	necomd 2837	. . . . . . . . . 10 ⊢ ((#‘{𝐵, 𝐶}) ≤ 2 → (#‘{𝐵, 𝐶}) ≠ 3)
61	60	adantl 481	. . . . . . . . 9 ⊢ (({𝐵, 𝐶} ∈ Fin ∧ (#‘{𝐵, 𝐶}) ≤ 2) → (#‘{𝐵, 𝐶}) ≠ 3)
62	42, 61	mp1i 13	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐵, 𝐶}) ≠ 3)
63		tpeq1 4221	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐵, 𝐶})
64		tpidm12 4234	. . . . . . . . . . 11 ⊢ {𝐵, 𝐵, 𝐶} = {𝐵, 𝐶}
65	63, 64	syl6req 2661	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐶} = {𝐴, 𝐵, 𝐶})
66	65	fveq2d 6107	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (#‘{𝐵, 𝐶}) = (#‘{𝐴, 𝐵, 𝐶}))
67	66	neeq1d 2841	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → ((#‘{𝐵, 𝐶}) ≠ 3 ↔ (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
68	62, 67	syl5ib 233	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
69	41, 68	sylbi 206	. . . . . 6 ⊢ (¬ 𝐴 ≠ 𝐵 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
70		hashprlei 13107	. . . . . . . . 9 ⊢ ({𝐴, 𝐶} ∈ Fin ∧ (#‘{𝐴, 𝐶}) ≤ 2)
71		prfi 8120	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐶} ∈ Fin
72		hashcl 13009	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐶} ∈ Fin → (#‘{𝐴, 𝐶}) ∈ ℕ₀)
73	72	nn0zd 11356	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (#‘{𝐴, 𝐶}) ∈ ℤ)
74	71, 73	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐴, 𝐶}) ∈ ℤ
75		zleltp1 11305	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐶}) ≤ 2 ↔ (#‘{𝐴, 𝐶}) < (2 + 1)))
76	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((#‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → (2 + 1) = 3)
77	76	breq2d 4595	. . . . . . . . . . . . . . 15 ⊢ (((#‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐶}) < (2 + 1) ↔ (#‘{𝐴, 𝐶}) < 3))
78	77	biimpd 218	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐶}) < (2 + 1) → (#‘{𝐴, 𝐶}) < 3))
79	75, 78	sylbid 229	. . . . . . . . . . . . 13 ⊢ (((#‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐶}) ≤ 2 → (#‘{𝐴, 𝐶}) < 3))
80	74, 47, 79	mp2an 704	. . . . . . . . . . . 12 ⊢ ((#‘{𝐴, 𝐶}) ≤ 2 → (#‘{𝐴, 𝐶}) < 3)
81	72	nn0red 11229	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (#‘{𝐴, 𝐶}) ∈ ℝ)
82	71, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐴, 𝐶}) ∈ ℝ
83	82, 57	ltnei 10040	. . . . . . . . . . . 12 ⊢ ((#‘{𝐴, 𝐶}) < 3 → 3 ≠ (#‘{𝐴, 𝐶}))
84	80, 83	syl 17	. . . . . . . . . . 11 ⊢ ((#‘{𝐴, 𝐶}) ≤ 2 → 3 ≠ (#‘{𝐴, 𝐶}))
85	84	necomd 2837	. . . . . . . . . 10 ⊢ ((#‘{𝐴, 𝐶}) ≤ 2 → (#‘{𝐴, 𝐶}) ≠ 3)
86	85	adantl 481	. . . . . . . . 9 ⊢ (({𝐴, 𝐶} ∈ Fin ∧ (#‘{𝐴, 𝐶}) ≤ 2) → (#‘{𝐴, 𝐶}) ≠ 3)
87	70, 86	mp1i 13	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐶}) ≠ 3)
88		tpeq2 4222	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐶})
89		tpidm23 4236	. . . . . . . . . . 11 ⊢ {𝐴, 𝐶, 𝐶} = {𝐴, 𝐶}
90	88, 89	syl6req 2661	. . . . . . . . . 10 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐶} = {𝐴, 𝐵, 𝐶})
91	90	fveq2d 6107	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (#‘{𝐴, 𝐶}) = (#‘{𝐴, 𝐵, 𝐶}))
92	91	neeq1d 2841	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → ((#‘{𝐴, 𝐶}) ≠ 3 ↔ (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
93	87, 92	syl5ib 233	. . . . . . 7 ⊢ (𝐵 = 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
94	6, 93	sylbi 206	. . . . . 6 ⊢ (¬ 𝐵 ≠ 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
95		hashprlei 13107	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ Fin ∧ (#‘{𝐴, 𝐵}) ≤ 2)
96		hashcl 13009	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ Fin → (#‘{𝐴, 𝐵}) ∈ ℕ₀)
97	96	nn0zd 11356	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (#‘{𝐴, 𝐵}) ∈ ℤ)
98	2, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐴, 𝐵}) ∈ ℤ
99		zleltp1 11305	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐵}) ≤ 2 ↔ (#‘{𝐴, 𝐵}) < (2 + 1)))
100	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((#‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → (2 + 1) = 3)
101	100	breq2d 4595	. . . . . . . . . . . . . . 15 ⊢ (((#‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐵}) < (2 + 1) ↔ (#‘{𝐴, 𝐵}) < 3))
102	101	biimpd 218	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐵}) < (2 + 1) → (#‘{𝐴, 𝐵}) < 3))
103	99, 102	sylbid 229	. . . . . . . . . . . . 13 ⊢ (((#‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐵}) ≤ 2 → (#‘{𝐴, 𝐵}) < 3))
104	98, 47, 103	mp2an 704	. . . . . . . . . . . 12 ⊢ ((#‘{𝐴, 𝐵}) ≤ 2 → (#‘{𝐴, 𝐵}) < 3)
105	96	nn0red 11229	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (#‘{𝐴, 𝐵}) ∈ ℝ)
106	2, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐴, 𝐵}) ∈ ℝ
107	106, 57	ltnei 10040	. . . . . . . . . . . 12 ⊢ ((#‘{𝐴, 𝐵}) < 3 → 3 ≠ (#‘{𝐴, 𝐵}))
108	104, 107	syl 17	. . . . . . . . . . 11 ⊢ ((#‘{𝐴, 𝐵}) ≤ 2 → 3 ≠ (#‘{𝐴, 𝐵}))
109	108	necomd 2837	. . . . . . . . . 10 ⊢ ((#‘{𝐴, 𝐵}) ≤ 2 → (#‘{𝐴, 𝐵}) ≠ 3)
110	109	adantl 481	. . . . . . . . 9 ⊢ (({𝐴, 𝐵} ∈ Fin ∧ (#‘{𝐴, 𝐵}) ≤ 2) → (#‘{𝐴, 𝐵}) ≠ 3)
111	95, 110	mp1i 13	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵}) ≠ 3)
112		tpeq3 4223	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐴})
113		tpidm13 4235	. . . . . . . . . . 11 ⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
114	112, 113	syl6req 2661	. . . . . . . . . 10 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐵, 𝐶})
115	114	fveq2d 6107	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (#‘{𝐴, 𝐵}) = (#‘{𝐴, 𝐵, 𝐶}))
116	115	neeq1d 2841	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → ((#‘{𝐴, 𝐵}) ≠ 3 ↔ (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
117	111, 116	syl5ib 233	. . . . . . 7 ⊢ (𝐶 = 𝐴 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
118	9, 117	sylbi 206	. . . . . 6 ⊢ (¬ 𝐶 ≠ 𝐴 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
119	69, 94, 118	3jaoi 1383	. . . . 5 ⊢ ((¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
120	21, 119	sylbi 206	. . . 4 ⊢ (¬ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
121	120	com12 32	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (¬ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
122	121	necon4bd 2802	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((#‘{𝐴, 𝐵, 𝐶}) = 3 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)))
123	40, 122	impbid 201	1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (#‘{𝐴, 𝐵, 𝐶}) = 3))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∪ cun 3538 {csn 4125 {cpr 4127 {ctp 4129 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 2c2 10947 3c3 10948 ℤcz 11254 #chash 12979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980

This theorem is referenced by: hashge3el3dif 13122 constr3lem2 26174 poimirlem9 32588 konigsberglem5 41426

W3C validator