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Theorem hashtpg 14408

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 9224	. . . . . . 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 4603	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
5		orcom 870	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐴))
6		nne 2936	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
7		eqcom 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
8	6, 7	bitr2i 276	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐵 ↔ ¬ 𝐵 ≠ 𝐶)
9		nne 2936	. . . . . . . . . . . . . . . . . . 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 14315	. . . . . . 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 14318	. . . . . . . 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 7373	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ((♯‘{𝐴, 𝐵}) + 1) = (2 + 1))
35	27, 34	eqtrd 2771	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘({𝐴, 𝐵} ∪ {𝐶})) = (2 + 1))
36		df-tp 4585	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
37	36	fveq2i 6837	. . . 4 ⊢ (♯‘{𝐴, 𝐵, 𝐶}) = (♯‘({𝐴, 𝐵} ∪ {𝐶}))
38		df-3 12209	. . . 4 ⊢ 3 = (2 + 1)
39	35, 37, 38	3eqtr4g 2796	. . 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 2936	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
42		hashprlei 14391	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ Fin ∧ (♯‘{𝐵, 𝐶}) ≤ 2)
43		prfi 9224	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐶} ∈ Fin
44		hashcl 14279	. . . . . . . . . . . . . . 15 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℕ₀)
45	44	nn0zd 12513	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℤ)
46	43, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℤ
47		2z 12523	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
48		zleltp1 12542	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) ≤ 2 ↔ (♯‘{𝐵, 𝐶}) < (2 + 1)))
49		2p1e3 12282	. . . . . . . . . . . . . . . . 17 ⊢ (2 + 1) = 3
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → (2 + 1) = 3)
51	50	breq2d 5110	. . . . . . . . . . . . . . 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 12463	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℝ)
56	43, 55	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℝ
57		3re 12225	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
58	56, 57	ltnei 11257	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐵, 𝐶}) < 3 → 3 ≠ (♯‘{𝐵, 𝐶}))
59	54, 58	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐵, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐵, 𝐶}))
60	59	necomd 2987	. . . . . . . . . 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 4699	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐵, 𝐶})
64		tpidm12 4712	. . . . . . . . . . 11 ⊢ {𝐵, 𝐵, 𝐶} = {𝐵, 𝐶}
65	63, 64	eqtr2di 2788	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐶} = {𝐴, 𝐵, 𝐶})
66	65	fveq2d 6838	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (♯‘{𝐵, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
67	66	neeq1d 2991	. . . . . . . 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 14391	. . . . . . . . 9 ⊢ ({𝐴, 𝐶} ∈ Fin ∧ (♯‘{𝐴, 𝐶}) ≤ 2)
71		prfi 9224	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐶} ∈ Fin
72		hashcl 14279	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℕ₀)
73	72	nn0zd 12513	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℤ)
74	71, 73	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℤ
75		zleltp1 12542	. . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . 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 12463	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℝ)
82	71, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℝ
83	82, 57	ltnei 11257	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐶}) < 3 → 3 ≠ (♯‘{𝐴, 𝐶}))
84	80, 83	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐶}))
85	84	necomd 2987	. . . . . . . . . 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 4700	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐶})
89		tpidm23 4714	. . . . . . . . . . 11 ⊢ {𝐴, 𝐶, 𝐶} = {𝐴, 𝐶}
90	88, 89	eqtr2di 2788	. . . . . . . . . 10 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐶} = {𝐴, 𝐵, 𝐶})
91	90	fveq2d 6838	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (♯‘{𝐴, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
92	91	neeq1d 2991	. . . . . . . 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 14391	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ Fin ∧ (♯‘{𝐴, 𝐵}) ≤ 2)
96		hashcl 14279	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℕ₀)
97	96	nn0zd 12513	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℤ)
98	2, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℤ
99		zleltp1 12542	. . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . 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 12463	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℝ)
106	2, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℝ
107	106, 57	ltnei 11257	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐵}) < 3 → 3 ≠ (♯‘{𝐴, 𝐵}))
108	104, 107	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐵}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐵}))
109	108	necomd 2987	. . . . . . . . . 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 4701	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐴})
113		tpidm13 4713	. . . . . . . . . . 11 ⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
114	112, 113	eqtr2di 2788	. . . . . . . . . 10 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐵, 𝐶})
115	114	fveq2d 6838	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (♯‘{𝐴, 𝐵}) = (♯‘{𝐴, 𝐵, 𝐶}))
116	115	neeq1d 2991	. . . . . . . 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 2952	. 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 1541 ∈ wcel 2113 ≠ wne 2932 ∪ cun 3899 {csn 4580 {cpr 4582 {ctp 4584 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 Fincfn 8883 ℝcr 11025 1c1 11027 + caddc 11029 < clt 11166 ≤ cle 11167 2c2 12200 3c3 12201 ℤcz 12488 ♯chash 14253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-xnn0 12475 df-z 12489 df-uz 12752 df-fz 13424 df-hash 14254

This theorem is referenced by: hash7g 14409 hashge3el3dif 14410 konigsberglem5 30331 poimirlem9 37830 usgrgrtrirex 48196 gpg3nbgrvtx0 48322 gpg3nbgrvtx0ALT 48323 gpg3nbgrvtx1 48324

W3C validator