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Theorem hashtpg 14127

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 1191	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → 𝐶 ∈ 𝑊)
2		prfi 9019	. . . . . . 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 4579	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
5		orcom 866	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐴))
6		nne 2946	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
7		eqcom 2745	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
8	6, 7	bitr2i 275	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐵 ↔ ¬ 𝐵 ≠ 𝐶)
9		nne 2946	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐶 ≠ 𝐴 ↔ 𝐶 = 𝐴)
10	9	bicomi 223	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐴 ↔ ¬ 𝐶 ≠ 𝐴)
11	8, 10	orbi12i 911	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐵 ∨ 𝐶 = 𝐴) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
12	5, 11	bitri 274	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
13	4, 12	bitrdi 286	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
14	13	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
15	14	3ad2ant3 1133	. . . . . . . . . . . . 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 870	. . . . . . . . . . 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 1088	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴) ↔ (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
20	18, 19	syl6ibr 251	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
21		3ianor 1105	. . . . . . . . 9 ⊢ (¬ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
22	20, 21	syl6ibr 251	. . . . . . . 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 14035	. . . . . . 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 833	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘({𝐴, 𝐵} ∪ {𝐶})) = ((♯‘{𝐴, 𝐵}) + 1))
28		simpr1 1192	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → 𝐴 ≠ 𝐵)
29		3simpa 1146	. . . . . . . . 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 14038	. . . . . . . 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 231	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘{𝐴, 𝐵}) = 2)
34	33	oveq1d 7270	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ((♯‘{𝐴, 𝐵}) + 1) = (2 + 1))
35	27, 34	eqtrd 2778	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (♯‘({𝐴, 𝐵} ∪ {𝐶})) = (2 + 1))
36		df-tp 4563	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
37	36	fveq2i 6759	. . . 4 ⊢ (♯‘{𝐴, 𝐵, 𝐶}) = (♯‘({𝐴, 𝐵} ∪ {𝐶}))
38		df-3 11967	. . . 4 ⊢ 3 = (2 + 1)
39	35, 37, 38	3eqtr4g 2804	. . 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 2946	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
42		hashprlei 14110	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ Fin ∧ (♯‘{𝐵, 𝐶}) ≤ 2)
43		prfi 9019	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐶} ∈ Fin
44		hashcl 13999	. . . . . . . . . . . . . . 15 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℕ₀)
45	44	nn0zd 12353	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℤ)
46	43, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℤ
47		2z 12282	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
48		zleltp1 12301	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) ≤ 2 ↔ (♯‘{𝐵, 𝐶}) < (2 + 1)))
49		2p1e3 12045	. . . . . . . . . . . . . . . . 17 ⊢ (2 + 1) = 3
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → (2 + 1) = 3)
51	50	breq2d 5082	. . . . . . . . . . . . . . 15 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) < (2 + 1) ↔ (♯‘{𝐵, 𝐶}) < 3))
52	51	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) < (2 + 1) → (♯‘{𝐵, 𝐶}) < 3))
53	48, 52	sylbid 239	. . . . . . . . . . . . 13 ⊢ (((♯‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐵, 𝐶}) ≤ 2 → (♯‘{𝐵, 𝐶}) < 3))
54	46, 47, 53	mp2an 688	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐵, 𝐶}) ≤ 2 → (♯‘{𝐵, 𝐶}) < 3)
55	44	nn0red 12224	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (♯‘{𝐵, 𝐶}) ∈ ℝ)
56	43, 55	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐵, 𝐶}) ∈ ℝ
57		3re 11983	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
58	56, 57	ltnei 11029	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐵, 𝐶}) < 3 → 3 ≠ (♯‘{𝐵, 𝐶}))
59	54, 58	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐵, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐵, 𝐶}))
60	59	necomd 2998	. . . . . . . . . 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 4675	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐵, 𝐶})
64		tpidm12 4688	. . . . . . . . . . 11 ⊢ {𝐵, 𝐵, 𝐶} = {𝐵, 𝐶}
65	63, 64	eqtr2di 2796	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐶} = {𝐴, 𝐵, 𝐶})
66	65	fveq2d 6760	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (♯‘{𝐵, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
67	66	neeq1d 3002	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → ((♯‘{𝐵, 𝐶}) ≠ 3 ↔ (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
68	62, 67	syl5ib 243	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
69	41, 68	sylbi 216	. . . . . 6 ⊢ (¬ 𝐴 ≠ 𝐵 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
70		hashprlei 14110	. . . . . . . . 9 ⊢ ({𝐴, 𝐶} ∈ Fin ∧ (♯‘{𝐴, 𝐶}) ≤ 2)
71		prfi 9019	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐶} ∈ Fin
72		hashcl 13999	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℕ₀)
73	72	nn0zd 12353	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℤ)
74	71, 73	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℤ
75		zleltp1 12301	. . . . . . . . . . . . . 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 5082	. . . . . . . . . . . . . . 15 ⊢ (((♯‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐶}) < (2 + 1) ↔ (♯‘{𝐴, 𝐶}) < 3))
78	77	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐶}) < (2 + 1) → (♯‘{𝐴, 𝐶}) < 3))
79	75, 78	sylbid 239	. . . . . . . . . . . . 13 ⊢ (((♯‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐶}) ≤ 2 → (♯‘{𝐴, 𝐶}) < 3))
80	74, 47, 79	mp2an 688	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐶}) ≤ 2 → (♯‘{𝐴, 𝐶}) < 3)
81	72	nn0red 12224	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (♯‘{𝐴, 𝐶}) ∈ ℝ)
82	71, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐶}) ∈ ℝ
83	82, 57	ltnei 11029	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐶}) < 3 → 3 ≠ (♯‘{𝐴, 𝐶}))
84	80, 83	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐶}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐶}))
85	84	necomd 2998	. . . . . . . . . 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 4676	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐶})
89		tpidm23 4690	. . . . . . . . . . 11 ⊢ {𝐴, 𝐶, 𝐶} = {𝐴, 𝐶}
90	88, 89	eqtr2di 2796	. . . . . . . . . 10 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐶} = {𝐴, 𝐵, 𝐶})
91	90	fveq2d 6760	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (♯‘{𝐴, 𝐶}) = (♯‘{𝐴, 𝐵, 𝐶}))
92	91	neeq1d 3002	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → ((♯‘{𝐴, 𝐶}) ≠ 3 ↔ (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
93	87, 92	syl5ib 243	. . . . . . 7 ⊢ (𝐵 = 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
94	6, 93	sylbi 216	. . . . . 6 ⊢ (¬ 𝐵 ≠ 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
95		hashprlei 14110	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ Fin ∧ (♯‘{𝐴, 𝐵}) ≤ 2)
96		hashcl 13999	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℕ₀)
97	96	nn0zd 12353	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℤ)
98	2, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℤ
99		zleltp1 12301	. . . . . . . . . . . . . 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 5082	. . . . . . . . . . . . . . 15 ⊢ (((♯‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐵}) < (2 + 1) ↔ (♯‘{𝐴, 𝐵}) < 3))
102	101	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (((♯‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐵}) < (2 + 1) → (♯‘{𝐴, 𝐵}) < 3))
103	99, 102	sylbid 239	. . . . . . . . . . . . 13 ⊢ (((♯‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((♯‘{𝐴, 𝐵}) ≤ 2 → (♯‘{𝐴, 𝐵}) < 3))
104	98, 47, 103	mp2an 688	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐵}) ≤ 2 → (♯‘{𝐴, 𝐵}) < 3)
105	96	nn0red 12224	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (♯‘{𝐴, 𝐵}) ∈ ℝ)
106	2, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘{𝐴, 𝐵}) ∈ ℝ
107	106, 57	ltnei 11029	. . . . . . . . . . . 12 ⊢ ((♯‘{𝐴, 𝐵}) < 3 → 3 ≠ (♯‘{𝐴, 𝐵}))
108	104, 107	syl 17	. . . . . . . . . . 11 ⊢ ((♯‘{𝐴, 𝐵}) ≤ 2 → 3 ≠ (♯‘{𝐴, 𝐵}))
109	108	necomd 2998	. . . . . . . . . 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 4677	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐴})
113		tpidm13 4689	. . . . . . . . . . 11 ⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
114	112, 113	eqtr2di 2796	. . . . . . . . . 10 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐵, 𝐶})
115	114	fveq2d 6760	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (♯‘{𝐴, 𝐵}) = (♯‘{𝐴, 𝐵, 𝐶}))
116	115	neeq1d 3002	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → ((♯‘{𝐴, 𝐵}) ≠ 3 ↔ (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
117	111, 116	syl5ib 243	. . . . . . 7 ⊢ (𝐶 = 𝐴 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
118	9, 117	sylbi 216	. . . . . 6 ⊢ (¬ 𝐶 ≠ 𝐴 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
119	69, 94, 118	3jaoi 1425	. . . . 5 ⊢ ((¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (♯‘{𝐴, 𝐵, 𝐶}) ≠ 3))
120	21, 119	sylbi 216	. . . 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 2962	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((♯‘{𝐴, 𝐵, 𝐶}) = 3 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)))
123	40, 122	impbid 211	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 205 ∧ wa 395 ∨ wo 843 ∨ w3o 1084 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∪ cun 3881 {csn 4558 {cpr 4560 {ctp 4562 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℝcr 10801 1c1 10803 + caddc 10805 < clt 10940 ≤ cle 10941 2c2 11958 3c3 11959 ℤcz 12249 ♯chash 13972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-fz 13169 df-hash 13973

This theorem is referenced by: hashge3el3dif 14128 konigsberglem5 28521 poimirlem9 35713

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