hashtpg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > hashtpg		Structured version Visualization version GIF version

Theorem hashtpg 13267

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 1066	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → 𝐶 ∈ 𝑊)
2		prfi 8235	. . . . . . 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 4196	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
5		orcom 402	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐴))
6		nne 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
7		eqcom 2629	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
8	6, 7	bitr2i 265	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐵 ↔ ¬ 𝐵 ≠ 𝐶)
9		nne 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝐶 ≠ 𝐴 ↔ 𝐶 = 𝐴)
10	9	bicomi 214	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝐴 ↔ ¬ 𝐶 ≠ 𝐴)
11	8, 10	orbi12i 543	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 = 𝐵 ∨ 𝐶 = 𝐴) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
12	5, 11	bitri 264	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
13	4, 12	syl6bb 276	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} ↔ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
14	13	biimpd 219	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
15	14	3ad2ant3 1084	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
16	15	imp 445	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝐶 ∈ {𝐴, 𝐵}) → (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
17	16	olcd 408	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝐶 ∈ {𝐴, 𝐵}) → (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
18	17	ex 450	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))))
19		3orass 1040	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴) ↔ (¬ 𝐴 ≠ 𝐵 ∨ (¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
20	18, 19	syl6ibr 242	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴)))
21		3ianor 1055	. . . . . . . . 9 ⊢ (¬ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴))
22	20, 21	syl6ibr 242	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ {𝐴, 𝐵} → ¬ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)))
23	22	con2d 129	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → ¬ 𝐶 ∈ {𝐴, 𝐵}))
24	23	imp 445	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ¬ 𝐶 ∈ {𝐴, 𝐵})
25		hashunsng 13181	. . . . . . 7 ⊢ (𝐶 ∈ 𝑊 → (({𝐴, 𝐵} ∈ Fin ∧ ¬ 𝐶 ∈ {𝐴, 𝐵}) → (#‘({𝐴, 𝐵} ∪ {𝐶})) = ((#‘{𝐴, 𝐵}) + 1)))
26	25	imp 445	. . . . . 6 ⊢ ((𝐶 ∈ 𝑊 ∧ ({𝐴, 𝐵} ∈ Fin ∧ ¬ 𝐶 ∈ {𝐴, 𝐵})) → (#‘({𝐴, 𝐵} ∪ {𝐶})) = ((#‘{𝐴, 𝐵}) + 1))
27	1, 3, 24, 26	syl12anc 1324	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (#‘({𝐴, 𝐵} ∪ {𝐶})) = ((#‘{𝐴, 𝐵}) + 1))
28		simpr1 1067	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → 𝐴 ≠ 𝐵)
29		3simpa 1058	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
30	29	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
31		hashprg 13182	. . . . . . . 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 222	. . . . . 6 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (#‘{𝐴, 𝐵}) = 2)
34	33	oveq1d 6665	. . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → ((#‘{𝐴, 𝐵}) + 1) = (2 + 1))
35	27, 34	eqtrd 2656	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (#‘({𝐴, 𝐵} ∪ {𝐶})) = (2 + 1))
36		df-tp 4182	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
37	36	fveq2i 6194	. . . 4 ⊢ (#‘{𝐴, 𝐵, 𝐶}) = (#‘({𝐴, 𝐵} ∪ {𝐶}))
38		df-3 11080	. . . 4 ⊢ 3 = (2 + 1)
39	35, 37, 38	3eqtr4g 2681	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)) → (#‘{𝐴, 𝐵, 𝐶}) = 3)
40	39	ex 450	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → (#‘{𝐴, 𝐵, 𝐶}) = 3))
41		nne 2798	. . . . . . 7 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)
42		hashprlei 13250	. . . . . . . . 9 ⊢ ({𝐵, 𝐶} ∈ Fin ∧ (#‘{𝐵, 𝐶}) ≤ 2)
43		prfi 8235	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐶} ∈ Fin
44		hashcl 13147	. . . . . . . . . . . . . . 15 ⊢ ({𝐵, 𝐶} ∈ Fin → (#‘{𝐵, 𝐶}) ∈ ℕ₀)
45	44	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (#‘{𝐵, 𝐶}) ∈ ℤ)
46	43, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐵, 𝐶}) ∈ ℤ
47		2z 11409	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
48		zleltp1 11428	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐵, 𝐶}) ≤ 2 ↔ (#‘{𝐵, 𝐶}) < (2 + 1)))
49		2p1e3 11151	. . . . . . . . . . . . . . . . 17 ⊢ (2 + 1) = 3
50	49	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → (2 + 1) = 3)
51	50	breq2d 4665	. . . . . . . . . . . . . . 15 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐵, 𝐶}) < (2 + 1) ↔ (#‘{𝐵, 𝐶}) < 3))
52	51	biimpd 219	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐵, 𝐶}) < (2 + 1) → (#‘{𝐵, 𝐶}) < 3))
53	48, 52	sylbid 230	. . . . . . . . . . . . 13 ⊢ (((#‘{𝐵, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐵, 𝐶}) ≤ 2 → (#‘{𝐵, 𝐶}) < 3))
54	46, 47, 53	mp2an 708	. . . . . . . . . . . 12 ⊢ ((#‘{𝐵, 𝐶}) ≤ 2 → (#‘{𝐵, 𝐶}) < 3)
55	44	nn0red 11352	. . . . . . . . . . . . . 14 ⊢ ({𝐵, 𝐶} ∈ Fin → (#‘{𝐵, 𝐶}) ∈ ℝ)
56	43, 55	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐵, 𝐶}) ∈ ℝ
57		3re 11094	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
58	56, 57	ltnei 10161	. . . . . . . . . . . 12 ⊢ ((#‘{𝐵, 𝐶}) < 3 → 3 ≠ (#‘{𝐵, 𝐶}))
59	54, 58	syl 17	. . . . . . . . . . 11 ⊢ ((#‘{𝐵, 𝐶}) ≤ 2 → 3 ≠ (#‘{𝐵, 𝐶}))
60	59	necomd 2849	. . . . . . . . . 10 ⊢ ((#‘{𝐵, 𝐶}) ≤ 2 → (#‘{𝐵, 𝐶}) ≠ 3)
61	60	adantl 482	. . . . . . . . 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 4277	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐵, 𝐵, 𝐶})
64		tpidm12 4290	. . . . . . . . . . 11 ⊢ {𝐵, 𝐵, 𝐶} = {𝐵, 𝐶}
65	63, 64	syl6req 2673	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → {𝐵, 𝐶} = {𝐴, 𝐵, 𝐶})
66	65	fveq2d 6195	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (#‘{𝐵, 𝐶}) = (#‘{𝐴, 𝐵, 𝐶}))
67	66	neeq1d 2853	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → ((#‘{𝐵, 𝐶}) ≠ 3 ↔ (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
68	62, 67	syl5ib 234	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
69	41, 68	sylbi 207	. . . . . 6 ⊢ (¬ 𝐴 ≠ 𝐵 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
70		hashprlei 13250	. . . . . . . . 9 ⊢ ({𝐴, 𝐶} ∈ Fin ∧ (#‘{𝐴, 𝐶}) ≤ 2)
71		prfi 8235	. . . . . . . . . . . . . 14 ⊢ {𝐴, 𝐶} ∈ Fin
72		hashcl 13147	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐶} ∈ Fin → (#‘{𝐴, 𝐶}) ∈ ℕ₀)
73	72	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (#‘{𝐴, 𝐶}) ∈ ℤ)
74	71, 73	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐴, 𝐶}) ∈ ℤ
75		zleltp1 11428	. . . . . . . . . . . . . 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 4665	. . . . . . . . . . . . . . 15 ⊢ (((#‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐶}) < (2 + 1) ↔ (#‘{𝐴, 𝐶}) < 3))
78	77	biimpd 219	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐶}) < (2 + 1) → (#‘{𝐴, 𝐶}) < 3))
79	75, 78	sylbid 230	. . . . . . . . . . . . 13 ⊢ (((#‘{𝐴, 𝐶}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐶}) ≤ 2 → (#‘{𝐴, 𝐶}) < 3))
80	74, 47, 79	mp2an 708	. . . . . . . . . . . 12 ⊢ ((#‘{𝐴, 𝐶}) ≤ 2 → (#‘{𝐴, 𝐶}) < 3)
81	72	nn0red 11352	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐶} ∈ Fin → (#‘{𝐴, 𝐶}) ∈ ℝ)
82	71, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐴, 𝐶}) ∈ ℝ
83	82, 57	ltnei 10161	. . . . . . . . . . . 12 ⊢ ((#‘{𝐴, 𝐶}) < 3 → 3 ≠ (#‘{𝐴, 𝐶}))
84	80, 83	syl 17	. . . . . . . . . . 11 ⊢ ((#‘{𝐴, 𝐶}) ≤ 2 → 3 ≠ (#‘{𝐴, 𝐶}))
85	84	necomd 2849	. . . . . . . . . 10 ⊢ ((#‘{𝐴, 𝐶}) ≤ 2 → (#‘{𝐴, 𝐶}) ≠ 3)
86	85	adantl 482	. . . . . . . . 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 4278	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐶})
89		tpidm23 4292	. . . . . . . . . . 11 ⊢ {𝐴, 𝐶, 𝐶} = {𝐴, 𝐶}
90	88, 89	syl6req 2673	. . . . . . . . . 10 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐶} = {𝐴, 𝐵, 𝐶})
91	90	fveq2d 6195	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (#‘{𝐴, 𝐶}) = (#‘{𝐴, 𝐵, 𝐶}))
92	91	neeq1d 2853	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → ((#‘{𝐴, 𝐶}) ≠ 3 ↔ (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
93	87, 92	syl5ib 234	. . . . . . 7 ⊢ (𝐵 = 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
94	6, 93	sylbi 207	. . . . . 6 ⊢ (¬ 𝐵 ≠ 𝐶 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
95		hashprlei 13250	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} ∈ Fin ∧ (#‘{𝐴, 𝐵}) ≤ 2)
96		hashcl 13147	. . . . . . . . . . . . . . 15 ⊢ ({𝐴, 𝐵} ∈ Fin → (#‘{𝐴, 𝐵}) ∈ ℕ₀)
97	96	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (#‘{𝐴, 𝐵}) ∈ ℤ)
98	2, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐴, 𝐵}) ∈ ℤ
99		zleltp1 11428	. . . . . . . . . . . . . 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 4665	. . . . . . . . . . . . . . 15 ⊢ (((#‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐵}) < (2 + 1) ↔ (#‘{𝐴, 𝐵}) < 3))
102	101	biimpd 219	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐵}) < (2 + 1) → (#‘{𝐴, 𝐵}) < 3))
103	99, 102	sylbid 230	. . . . . . . . . . . . 13 ⊢ (((#‘{𝐴, 𝐵}) ∈ ℤ ∧ 2 ∈ ℤ) → ((#‘{𝐴, 𝐵}) ≤ 2 → (#‘{𝐴, 𝐵}) < 3))
104	98, 47, 103	mp2an 708	. . . . . . . . . . . 12 ⊢ ((#‘{𝐴, 𝐵}) ≤ 2 → (#‘{𝐴, 𝐵}) < 3)
105	96	nn0red 11352	. . . . . . . . . . . . . 14 ⊢ ({𝐴, 𝐵} ∈ Fin → (#‘{𝐴, 𝐵}) ∈ ℝ)
106	2, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (#‘{𝐴, 𝐵}) ∈ ℝ
107	106, 57	ltnei 10161	. . . . . . . . . . . 12 ⊢ ((#‘{𝐴, 𝐵}) < 3 → 3 ≠ (#‘{𝐴, 𝐵}))
108	104, 107	syl 17	. . . . . . . . . . 11 ⊢ ((#‘{𝐴, 𝐵}) ≤ 2 → 3 ≠ (#‘{𝐴, 𝐵}))
109	108	necomd 2849	. . . . . . . . . 10 ⊢ ((#‘{𝐴, 𝐵}) ≤ 2 → (#‘{𝐴, 𝐵}) ≠ 3)
110	109	adantl 482	. . . . . . . . 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 4279	. . . . . . . . . . 11 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐴})
113		tpidm13 4291	. . . . . . . . . . 11 ⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
114	112, 113	syl6req 2673	. . . . . . . . . 10 ⊢ (𝐶 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐵, 𝐶})
115	114	fveq2d 6195	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (#‘{𝐴, 𝐵}) = (#‘{𝐴, 𝐵, 𝐶}))
116	115	neeq1d 2853	. . . . . . . 8 ⊢ (𝐶 = 𝐴 → ((#‘{𝐴, 𝐵}) ≠ 3 ↔ (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
117	111, 116	syl5ib 234	. . . . . . 7 ⊢ (𝐶 = 𝐴 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
118	9, 117	sylbi 207	. . . . . 6 ⊢ (¬ 𝐶 ≠ 𝐴 → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
119	69, 94, 118	3jaoi 1391	. . . . 5 ⊢ ((¬ 𝐴 ≠ 𝐵 ∨ ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐶 ≠ 𝐴) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (#‘{𝐴, 𝐵, 𝐶}) ≠ 3))
120	21, 119	sylbi 207	. . . 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 2814	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((#‘{𝐴, 𝐵, 𝐶}) = 3 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴)))
123	40, 122	impbid 202	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 196 ∨ wo 383 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∪ cun 3572 {csn 4177 {cpr 4179 {ctp 4181 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℝcr 9935 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 2c2 11070 3c3 11071 ℤcz 11377 #chash 13117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118

This theorem is referenced by: hashge3el3dif 13268 konigsberglem5 27118 poimirlem9 33418

W3C validator