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Theorem hashtpg 14457

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 1194	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → 𝐶 ∈ 𝑊)
2		prfi 9281	. . . . . . 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 4615	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
5		orcom 870	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐴))
6		nne 2930	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
7		eqcom 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
8	6, 7	bitr2i 276	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐵 ↔ ¬ 𝐵 ≠ 𝐶)
9		nne 2930	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐶 ≠ 𝐴 ↔ 𝐶 = 𝐴)
10	9	bicomi 224	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐴 ↔ ¬ 𝐶 ≠ 𝐴)
11	8, 10	orbi12i 914	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐵 ∨ 𝐶 = 𝐴) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
12	5, 11	bitri 275	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
13	4, 12	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
14	13	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
15	14	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
16	15	imp 406	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝐶 ∈ {𝐴, 𝐵}) → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
17	16	olcd 874	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝐶 ∈ {𝐴, 𝐵}) → (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
18	17	ex 412	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))))
19		3orass 1089	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴) ↔ (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
20	18, 19	imbitrrdi 252	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
21		3ianor 1106	. . . . . . . . 9 ⊢ (¬ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
22	20, 21	imbitrrdi 252	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → ¬ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)))
23	22	con2d 134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → ¬ 𝐶 ∈ {𝐴, 𝐵}))
24	23	imp 406	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ¬ 𝐶 ∈ {𝐴, 𝐵})
25		hashunsng 14364	. . . . . . 7 ⊢ (𝐶 ∈ 𝑊 → (({𝐴, 𝐵} ∈ Fin ∧ ¬ 𝐶 ∈ {𝐴, 𝐵}) → (♯‘({𝐴, 𝐵} ∪ {𝐶})) = ((♯‘{𝐴, 𝐵}) + 1)))
26	25	imp 406	. . . . . 6 ⊢ ((𝐶 ∈ 𝑊 ∧ ({𝐴, 𝐵} ∈ Fin ∧ ¬ 𝐶 ∈ {𝐴, 𝐵})) → (♯‘({𝐴, 𝐵} ∪ {𝐶})) = ((♯‘{𝐴, 𝐵}) + 1))
27	1, 3, 24, 26	syl12anc 836	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘({𝐴, 𝐵} ∪ {𝐶})) = ((♯‘{𝐴, 𝐵}) + 1))
28		simpr1 1195	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → 𝐴 ≠ 𝐵)
29		3simpa 1148	. . . . . . . . 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 14367	. . . . . . . 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 232	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘{𝐴, 𝐵}) = 2)
34	33	oveq1d 7405	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ((♯‘{𝐴, 𝐵}) + 1) = (2 + 1))
35	27, 34	eqtrd 2765	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘({𝐴, 𝐵} ∪ {𝐶})) = (2 + 1))
36		df-tp 4597	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
37	36	fveq2i 6864	. . . 4 ⊢ (♯‘{𝐴, 𝐵, 𝐶}) = (♯‘({𝐴, 𝐵} ∪ {𝐶}))
38		df-3 12257	. . . 4 ⊢ 3 = (2 + 1)
39	35, 37, 38	3eqtr4g 2790	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘{𝐴, 𝐵, 𝐶}) = 3)
40	39	ex 412	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → (♯‘{𝐴, 𝐵, 𝐶}) = 3))
41		nne 2930	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
42		hashprlei 14440	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ Fin ∧ (♯‘{𝐵, 𝐶}) ≤ 2)
43		prfi 9281	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐶} ∈ Fin
44		hashcl 14328	. . . . . . . . . . . . . . 15 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℕ₀)
45	44	nn0zd 12562	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℤ)
46	43, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℤ
47		2z 12572	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
48		zleltp1 12591	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) ≤ 2 ↔ (♯‘{𝐵, 𝐶}) < (2 + 1)))
49		2p1e3 12330	. . . . . . . . . . . . . . . . 17 ⊢ (2 + 1) = 3
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → (2 + 1) = 3)
51	50	breq2d 5122	. . . . . . . . . . . . . . 15 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) < (2 + 1) ↔ (♯‘{𝐵, 𝐶}) < 3))
52	51	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) < (2 + 1) → (♯‘{𝐵, 𝐶}) < 3))
53	48, 52	sylbid 240	. . . . . . . . . . . . 13 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) ≤ 2 → (♯‘{𝐵, 𝐶}) < 3))
54	46, 47, 53	mp2an 692	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐵, 𝐶}) ≤ 2 → (♯‘{𝐵, 𝐶}) < 3)
55	44	nn0red 12511	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℝ)
56	43, 55	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℝ
57		3re 12273	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
58	56, 57	ltnei 11305	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐵, 𝐶}) < 3 → 3 ≠ (♯‘{𝐵, 𝐶}))
59	54, 58	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐵, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐵, 𝐶}))
60	59	necomd 2981	. . . . . . . . . 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 4709	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐵, 𝐶})
64		tpidm12 4722	. . . . . . . . . . 11 ⊢ {𝐵, 𝐵, 𝐶} = {𝐵, 𝐶}
65	63, 64	eqtr2di 2782	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐶} = {𝐴, 𝐵, 𝐶})
66	65	fveq2d 6865	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (♯‘{𝐵, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
67	66	neeq1d 2985	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → ((♯‘{𝐵, 𝐶}) ≠ 3 ↔ (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
68	62, 67	imbitrid 244	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
69	41, 68	sylbi 217	. . . . . 6 ⊢ (¬ 𝐴 ≠ 𝐵 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
70		hashprlei 14440	. . . . . . . . 9 ⊢ ({𝐴, 𝐶} ∈ Fin ∧ (♯‘{𝐴, 𝐶}) ≤ 2)
71		prfi 9281	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐶} ∈ Fin
72		hashcl 14328	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℕ₀)
73	72	nn0zd 12562	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℤ)
74	71, 73	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℤ
75		zleltp1 12591	. . . . . . . . . . . . . 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 5122	. . . . . . . . . . . . . . 15 ⊢ (((♯‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐶}) < (2 + 1) ↔ (♯‘{𝐴, 𝐶}) < 3))
78	77	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐶}) < (2 + 1) → (♯‘{𝐴, 𝐶}) < 3))
79	75, 78	sylbid 240	. . . . . . . . . . . . 13 ⊢ (((♯‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐶}) ≤ 2 → (♯‘{𝐴, 𝐶}) < 3))
80	74, 47, 79	mp2an 692	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐶}) ≤ 2 → (♯‘{𝐴, 𝐶}) < 3)
81	72	nn0red 12511	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℝ)
82	71, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℝ
83	82, 57	ltnei 11305	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐶}) < 3 → 3 ≠ (♯‘{𝐴, 𝐶}))
84	80, 83	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐶}))
85	84	necomd 2981	. . . . . . . . . 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 4710	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐶})
89		tpidm23 4724	. . . . . . . . . . 11 ⊢ {𝐴, 𝐶, 𝐶} = {𝐴, 𝐶}
90	88, 89	eqtr2di 2782	. . . . . . . . . 10 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐶} = {𝐴, 𝐵, 𝐶})
91	90	fveq2d 6865	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (♯‘{𝐴, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
92	91	neeq1d 2985	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → ((♯‘{𝐴, 𝐶}) ≠ 3 ↔ (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
93	87, 92	imbitrid 244	. . . . . . 7 ⊢ (𝐵 = 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
94	6, 93	sylbi 217	. . . . . 6 ⊢ (¬ 𝐵 ≠ 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
95		hashprlei 14440	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ Fin ∧ (♯‘{𝐴, 𝐵}) ≤ 2)
96		hashcl 14328	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℕ₀)
97	96	nn0zd 12562	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℤ)
98	2, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℤ
99		zleltp1 12591	. . . . . . . . . . . . . 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 5122	. . . . . . . . . . . . . . 15 ⊢ (((♯‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐵}) < (2 + 1) ↔ (♯‘{𝐴, 𝐵}) < 3))
102	101	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐵}) < (2 + 1) → (♯‘{𝐴, 𝐵}) < 3))
103	99, 102	sylbid 240	. . . . . . . . . . . . 13 ⊢ (((♯‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐵}) ≤ 2 → (♯‘{𝐴, 𝐵}) < 3))
104	98, 47, 103	mp2an 692	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐵}) ≤ 2 → (♯‘{𝐴, 𝐵}) < 3)
105	96	nn0red 12511	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℝ)
106	2, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℝ
107	106, 57	ltnei 11305	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐵}) < 3 → 3 ≠ (♯‘{𝐴, 𝐵}))
108	104, 107	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐵}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐵}))
109	108	necomd 2981	. . . . . . . . . 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 4711	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐴})
113		tpidm13 4723	. . . . . . . . . . 11 ⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
114	112, 113	eqtr2di 2782	. . . . . . . . . 10 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐵, 𝐶})
115	114	fveq2d 6865	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (♯‘{𝐴, 𝐵}) = (♯‘{𝐴, 𝐵, 𝐶}))
116	115	neeq1d 2985	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → ((♯‘{𝐴, 𝐵}) ≠ 3 ↔ (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
117	111, 116	imbitrid 244	. . . . . . 7 ⊢ (𝐶 = 𝐴 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
118	9, 117	sylbi 217	. . . . . 6 ⊢ (¬ 𝐶 ≠ 𝐴 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
119	69, 94, 118	3jaoi 1430	. . . . 5 ⊢ ((¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
120	21, 119	sylbi 217	. . . 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 2946	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((♯‘{𝐴, 𝐵, 𝐶}) = 3 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)))
123	40, 122	impbid 212	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 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∪ cun 3915 {csn 4592 {cpr 4594 {ctp 4596 class class class wbr 5110 ‘cfv 6514 (class class class)co 7390 Fincfn 8921 ℝcr 11074 1c1 11076 + caddc 11078 < clt 11215 ≤ cle 11216 2c2 12248 3c3 12249 ℤcz 12536 ♯chash 14302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-xnn0 12523 df-z 12537 df-uz 12801 df-fz 13476 df-hash 14303

This theorem is referenced by: hash7g 14458 hashge3el3dif 14459 konigsberglem5 30192 poimirlem9 37630 usgrgrtrirex 47953 gpg3nbgrvtx0 48071 gpg3nbgrvtx0ALT 48072 gpg3nbgrvtx1 48073

W3C validator