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Theorem hashtpg 14399

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 9219	. . . . . . 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 4600	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
5		orcom 870	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐴))
6		nne 2933	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
7		eqcom 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
8	6, 7	bitr2i 276	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐵 ↔ ¬ 𝐵 ≠ 𝐶)
9		nne 2933	. . . . . . . . . . . . . . . . . . 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 14306	. . . . . . 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 14309	. . . . . . . 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 7370	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ((♯‘{𝐴, 𝐵}) + 1) = (2 + 1))
35	27, 34	eqtrd 2768	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘({𝐴, 𝐵} ∪ {𝐶})) = (2 + 1))
36		df-tp 4582	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
37	36	fveq2i 6834	. . . 4 ⊢ (♯‘{𝐴, 𝐵, 𝐶}) = (♯‘({𝐴, 𝐵} ∪ {𝐶}))
38		df-3 12200	. . . 4 ⊢ 3 = (2 + 1)
39	35, 37, 38	3eqtr4g 2793	. . 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 2933	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
42		hashprlei 14382	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ Fin ∧ (♯‘{𝐵, 𝐶}) ≤ 2)
43		prfi 9219	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐶} ∈ Fin
44		hashcl 14270	. . . . . . . . . . . . . . 15 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℕ₀)
45	44	nn0zd 12504	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℤ)
46	43, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℤ
47		2z 12514	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
48		zleltp1 12533	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) ≤ 2 ↔ (♯‘{𝐵, 𝐶}) < (2 + 1)))
49		2p1e3 12273	. . . . . . . . . . . . . . . . 17 ⊢ (2 + 1) = 3
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → (2 + 1) = 3)
51	50	breq2d 5107	. . . . . . . . . . . . . . 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 12454	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℝ)
56	43, 55	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℝ
57		3re 12216	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
58	56, 57	ltnei 11248	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐵, 𝐶}) < 3 → 3 ≠ (♯‘{𝐵, 𝐶}))
59	54, 58	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐵, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐵, 𝐶}))
60	59	necomd 2984	. . . . . . . . . 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 4696	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐵, 𝐶})
64		tpidm12 4709	. . . . . . . . . . 11 ⊢ {𝐵, 𝐵, 𝐶} = {𝐵, 𝐶}
65	63, 64	eqtr2di 2785	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐶} = {𝐴, 𝐵, 𝐶})
66	65	fveq2d 6835	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (♯‘{𝐵, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
67	66	neeq1d 2988	. . . . . . . 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 14382	. . . . . . . . 9 ⊢ ({𝐴, 𝐶} ∈ Fin ∧ (♯‘{𝐴, 𝐶}) ≤ 2)
71		prfi 9219	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐶} ∈ Fin
72		hashcl 14270	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℕ₀)
73	72	nn0zd 12504	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℤ)
74	71, 73	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℤ
75		zleltp1 12533	. . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . . 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 12454	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℝ)
82	71, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℝ
83	82, 57	ltnei 11248	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐶}) < 3 → 3 ≠ (♯‘{𝐴, 𝐶}))
84	80, 83	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐶}))
85	84	necomd 2984	. . . . . . . . . 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 4697	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐶})
89		tpidm23 4711	. . . . . . . . . . 11 ⊢ {𝐴, 𝐶, 𝐶} = {𝐴, 𝐶}
90	88, 89	eqtr2di 2785	. . . . . . . . . 10 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐶} = {𝐴, 𝐵, 𝐶})
91	90	fveq2d 6835	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (♯‘{𝐴, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
92	91	neeq1d 2988	. . . . . . . 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 14382	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ Fin ∧ (♯‘{𝐴, 𝐵}) ≤ 2)
96		hashcl 14270	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℕ₀)
97	96	nn0zd 12504	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℤ)
98	2, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℤ
99		zleltp1 12533	. . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . . 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 12454	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℝ)
106	2, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℝ
107	106, 57	ltnei 11248	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐵}) < 3 → 3 ≠ (♯‘{𝐴, 𝐵}))
108	104, 107	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐵}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐵}))
109	108	necomd 2984	. . . . . . . . . 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 4698	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐴})
113		tpidm13 4710	. . . . . . . . . . 11 ⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
114	112, 113	eqtr2di 2785	. . . . . . . . . 10 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐵, 𝐶})
115	114	fveq2d 6835	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (♯‘{𝐴, 𝐵}) = (♯‘{𝐴, 𝐵, 𝐶}))
116	115	neeq1d 2988	. . . . . . . 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 2949	. 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 2929 ∪ cun 3896 {csn 4577 {cpr 4579 {ctp 4581 class class class wbr 5095 ‘cfv 6489 (class class class)co 7355 Fincfn 8879 ℝcr 11016 1c1 11018 + caddc 11020 < clt 11157 ≤ cle 11158 2c2 12191 3c3 12192 ℤcz 12479 ♯chash 14244

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 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-er 8631 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-dju 9805 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-3 12200 df-n0 12393 df-xnn0 12466 df-z 12480 df-uz 12743 df-fz 13415 df-hash 14245

This theorem is referenced by: hash7g 14400 hashge3el3dif 14401 konigsberglem5 30257 poimirlem9 37742 usgrgrtrirex 48112 gpg3nbgrvtx0 48238 gpg3nbgrvtx0ALT 48239 gpg3nbgrvtx1 48240

W3C validator