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Theorem hashtpg 14534

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 1193	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → 𝐶 ∈ 𝑊)
2		prfi 9391	. . . . . . 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 4670	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
5		orcom 869	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐴))
6		nne 2950	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
7		eqcom 2747	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
8	6, 7	bitr2i 276	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐵 ↔ ¬ 𝐵 ≠ 𝐶)
9		nne 2950	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐶 ≠ 𝐴 ↔ 𝐶 = 𝐴)
10	9	bicomi 224	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐴 ↔ ¬ 𝐶 ≠ 𝐴)
11	8, 10	orbi12i 913	. . . . . . . . . . . . . . . . 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 873	. . . . . . . . . . 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 1090	. . . . . . . . . 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 1107	. . . . . . . . 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 14441	. . . . . . 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 1194	. . . . . . 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 14444	. . . . . . . 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 7463	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ((♯‘{𝐴, 𝐵}) + 1) = (2 + 1))
35	27, 34	eqtrd 2780	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘({𝐴, 𝐵} ∪ {𝐶})) = (2 + 1))
36		df-tp 4653	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
37	36	fveq2i 6923	. . . 4 ⊢ (♯‘{𝐴, 𝐵, 𝐶}) = (♯‘({𝐴, 𝐵} ∪ {𝐶}))
38		df-3 12357	. . . 4 ⊢ 3 = (2 + 1)
39	35, 37, 38	3eqtr4g 2805	. . 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 2950	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
42		hashprlei 14517	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ Fin ∧ (♯‘{𝐵, 𝐶}) ≤ 2)
43		prfi 9391	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐶} ∈ Fin
44		hashcl 14405	. . . . . . . . . . . . . . 15 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℕ₀)
45	44	nn0zd 12665	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℤ)
46	43, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℤ
47		2z 12675	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
48		zleltp1 12694	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) ≤ 2 ↔ (♯‘{𝐵, 𝐶}) < (2 + 1)))
49		2p1e3 12435	. . . . . . . . . . . . . . . . 17 ⊢ (2 + 1) = 3
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → (2 + 1) = 3)
51	50	breq2d 5178	. . . . . . . . . . . . . . 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 691	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐵, 𝐶}) ≤ 2 → (♯‘{𝐵, 𝐶}) < 3)
55	44	nn0red 12614	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℝ)
56	43, 55	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℝ
57		3re 12373	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
58	56, 57	ltnei 11414	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐵, 𝐶}) < 3 → 3 ≠ (♯‘{𝐵, 𝐶}))
59	54, 58	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐵, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐵, 𝐶}))
60	59	necomd 3002	. . . . . . . . . 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 4767	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐵, 𝐶})
64		tpidm12 4780	. . . . . . . . . . 11 ⊢ {𝐵, 𝐵, 𝐶} = {𝐵, 𝐶}
65	63, 64	eqtr2di 2797	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐶} = {𝐴, 𝐵, 𝐶})
66	65	fveq2d 6924	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (♯‘{𝐵, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
67	66	neeq1d 3006	. . . . . . . 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 14517	. . . . . . . . 9 ⊢ ({𝐴, 𝐶} ∈ Fin ∧ (♯‘{𝐴, 𝐶}) ≤ 2)
71		prfi 9391	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐶} ∈ Fin
72		hashcl 14405	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℕ₀)
73	72	nn0zd 12665	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℤ)
74	71, 73	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℤ
75		zleltp1 12694	. . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . 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 691	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐶}) ≤ 2 → (♯‘{𝐴, 𝐶}) < 3)
81	72	nn0red 12614	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℝ)
82	71, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℝ
83	82, 57	ltnei 11414	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐶}) < 3 → 3 ≠ (♯‘{𝐴, 𝐶}))
84	80, 83	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐶}))
85	84	necomd 3002	. . . . . . . . . 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 4768	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐶})
89		tpidm23 4782	. . . . . . . . . . 11 ⊢ {𝐴, 𝐶, 𝐶} = {𝐴, 𝐶}
90	88, 89	eqtr2di 2797	. . . . . . . . . 10 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐶} = {𝐴, 𝐵, 𝐶})
91	90	fveq2d 6924	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (♯‘{𝐴, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
92	91	neeq1d 3006	. . . . . . . 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 14517	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ Fin ∧ (♯‘{𝐴, 𝐵}) ≤ 2)
96		hashcl 14405	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℕ₀)
97	96	nn0zd 12665	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℤ)
98	2, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℤ
99		zleltp1 12694	. . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . 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 691	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐵}) ≤ 2 → (♯‘{𝐴, 𝐵}) < 3)
105	96	nn0red 12614	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℝ)
106	2, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℝ
107	106, 57	ltnei 11414	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐵}) < 3 → 3 ≠ (♯‘{𝐴, 𝐵}))
108	104, 107	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐵}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐵}))
109	108	necomd 3002	. . . . . . . . . 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 4769	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐴})
113		tpidm13 4781	. . . . . . . . . . 11 ⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
114	112, 113	eqtr2di 2797	. . . . . . . . . 10 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐵, 𝐶})
115	114	fveq2d 6924	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (♯‘{𝐴, 𝐵}) = (♯‘{𝐴, 𝐵, 𝐶}))
116	115	neeq1d 3006	. . . . . . . 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 1428	. . . . 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 2966	. 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 846 ∨ w3o 1086 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∪ cun 3974 {csn 4648 {cpr 4650 {ctp 4652 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 Fincfn 9003 ℝcr 11183 1c1 11185 + caddc 11187 < clt 11324 ≤ cle 11325 2c2 12348 3c3 12349 ℤcz 12639 ♯chash 14379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-fz 13568 df-hash 14380

This theorem is referenced by: hash7g 14535 hashge3el3dif 14536 konigsberglem5 30288 poimirlem9 37589 usgrgrtrirex 47799

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