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Theorem hashgt23el 14457

Description: A set with more than two elements has at least three different elements. (Contributed by BTernaryTau, 21-Sep-2023.)

**Assertion**
Ref	Expression
hashgt23el	⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))

Distinct variable groups: 𝑊,𝑎 𝑉,𝑎,𝑏,𝑐 Allowed substitution hints: 𝑊(𝑏,𝑐)

**Proof of Theorem hashgt23el**
Step	Hyp	Ref	Expression
1		2pos 12340	. . . . . 6 ⊢ 0 < 2
2		0xr 11251	. . . . . . 7 ⊢ 0 ∈ ℝ^*
3		2re 12310	. . . . . . . 8 ⊢ 2 ∈ ℝ
4	3	rexri 11262	. . . . . . 7 ⊢ 2 ∈ ℝ^*
5		hashxrcl 14389	. . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → (♯‘𝑉) ∈ ℝ^*)
6		xrlttr 13160	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 2 ∈ ℝ^* ∧ (♯‘𝑉) ∈ ℝ^*) → ((0 < 2 ∧ 2 < (♯‘𝑉)) → 0 < (♯‘𝑉)))
7	2, 4, 5, 6	mp3an12i 1494	. . . . . 6 ⊢ (𝑉 ∈ 𝑊 → ((0 < 2 ∧ 2 < (♯‘𝑉)) → 0 < (♯‘𝑉)))
8	1, 7	mpani 708	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → (2 < (♯‘𝑉) → 0 < (♯‘𝑉)))
9		hashgt0elex 14433	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 0 < (♯‘𝑉)) → ∃𝑎 𝑎 ∈ 𝑉)
10	9	ex 417	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → (0 < (♯‘𝑉) → ∃𝑎 𝑎 ∈ 𝑉))
11	8, 10	syld 48	. . . 4 ⊢ (𝑉 ∈ 𝑊 → (2 < (♯‘𝑉) → ∃𝑎 𝑎 ∈ 𝑉))
12	11	imp 411	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎 𝑎 ∈ 𝑉)
13		difexg 5300	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → (𝑉 ∖ {𝑎}) ∈ V)
14		difsnid 4776	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑉 → ((𝑉 ∖ {𝑎}) ∪ {𝑎}) = 𝑉)
15	14	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝑉 → (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) = (♯‘𝑉))
16	15	breq2d 5121	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝑉 → (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) ↔ 2 < (♯‘𝑉)))
17	16	adantr 485	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊) → (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) ↔ 2 < (♯‘𝑉)))
18		df-2 12298	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
19	18	breq1i 5116	. . . . . . . . . . . 12 ⊢ (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})))
20		neldifsn 4760	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑎 ∈ (𝑉 ∖ {𝑎})
21		1nn0 12515	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
22		hashunsnggt 14426	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑉 ∖ {𝑎}) ∈ V ∧ 𝑎 ∈ 𝑉 ∧ 1 ∈ ℕ₀) ∧ ¬ 𝑎 ∈ (𝑉 ∖ {𝑎})) → (1 < (♯‘(𝑉 ∖ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))))
23	21, 22	mp3anl3 1486	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 ∖ {𝑎}) ∈ V ∧ 𝑎 ∈ 𝑉) ∧ ¬ 𝑎 ∈ (𝑉 ∖ {𝑎})) → (1 < (♯‘(𝑉 ∖ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))))
24	13, 23	sylanl1 692	. . . . . . . . . . . . . 14 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉) ∧ ¬ 𝑎 ∈ (𝑉 ∖ {𝑎})) → (1 < (♯‘(𝑉 ∖ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))))
25	20, 24	mpan2 703	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉) → (1 < (♯‘(𝑉 ∖ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))))
26	25	biimp3ar 1499	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉 ∧ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))) → 1 < (♯‘(𝑉 ∖ {𝑎})))
27	19, 26	syl3an3b 1432	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉 ∧ 2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))) → 1 < (♯‘(𝑉 ∖ {𝑎})))
28	27	3expia 1139	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉) → (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
29	28	ancoms 463	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊) → (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
30	17, 29	sylbird 263	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊) → (2 < (♯‘𝑉) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
31	30	3impia 1135	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎})))
32	31	3expib 1140	. . . . . 6 ⊢ (𝑎 ∈ 𝑉 → ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
33		1lt2 12408	. . . . . . . . . . 11 ⊢ 1 < 2
34		1xr 11263	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ^*
35		xrlttr 13160	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ^* ∧ 2 ∈ ℝ^* ∧ (♯‘𝑉) ∈ ℝ^*) → ((1 < 2 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘𝑉)))
36	34, 4, 5, 35	mp3an12i 1494	. . . . . . . . . . 11 ⊢ (𝑉 ∈ 𝑊 → ((1 < 2 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘𝑉)))
37	33, 36	mpani 708	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝑊 → (2 < (♯‘𝑉) → 1 < (♯‘𝑉)))
38	37	imp 411	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘𝑉))
39	38	3adant1 1148	. . . . . . . 8 ⊢ ((¬ 𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘𝑉))
40		difsn 4766	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ 𝑉 → (𝑉 ∖ {𝑎}) = 𝑉)
41	40	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((¬ 𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → (𝑉 ∖ {𝑎}) = 𝑉)
42	41	fveq2d 6885	. . . . . . . 8 ⊢ ((¬ 𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → (♯‘(𝑉 ∖ {𝑎})) = (♯‘𝑉))
43	39, 42	breqtrrd 5139	. . . . . . 7 ⊢ ((¬ 𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎})))
44	43	3expib 1140	. . . . . 6 ⊢ (¬ 𝑎 ∈ 𝑉 → ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
45	32, 44	pm2.61i 184	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎})))
46		hashgt12el 14455	. . . . 5 ⊢ (((𝑉 ∖ {𝑎}) ∈ V ∧ 1 < (♯‘(𝑉 ∖ {𝑎}))) → ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐)
47	13, 45, 46	syl2an2r 697	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐)
48	47	alrimiv 1957	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∀𝑎∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐)
49		19.29r 1904	. . 3 ⊢ ((∃𝑎 𝑎 ∈ 𝑉 ∧ ∀𝑎∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐) → ∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐))
50	12, 48, 49	syl2anc 595	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐))
51		df-rex 3090	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 ↔ ∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐))
52		eldifsn 4753	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑏 ∈ 𝑉 ∧ 𝑏 ≠ 𝑎))
53		necom 3011	. . . . . . . . . 10 ⊢ (𝑏 ≠ 𝑎 ↔ 𝑎 ≠ 𝑏)
54	53	anbi2i 634	. . . . . . . . 9 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑏 ≠ 𝑎) ↔ (𝑏 ∈ 𝑉 ∧ 𝑎 ≠ 𝑏))
55	52, 54	bitri 278	. . . . . . . 8 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑏 ∈ 𝑉 ∧ 𝑎 ≠ 𝑏))
56		ax-5 1940	. . . . . . . . 9 ⊢ (𝑎 ≠ 𝑏 → ∀𝑐 𝑎 ≠ 𝑏)
57	56	anim2i 628	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑎 ≠ 𝑏) → (𝑏 ∈ 𝑉 ∧ ∀𝑐 𝑎 ≠ 𝑏))
58	55, 57	sylbi 220	. . . . . . 7 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → (𝑏 ∈ 𝑉 ∧ ∀𝑐 𝑎 ≠ 𝑏))
59		3anass 1111	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ (𝑐 ∈ 𝑉 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
60	59	exbii 1878	. . . . . . . . 9 ⊢ (∃𝑐(𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑐(𝑐 ∈ 𝑉 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
61		df-rex 3090	. . . . . . . . . 10 ⊢ (∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 ↔ ∃𝑐(𝑐 ∈ (𝑉 ∖ {𝑎}) ∧ 𝑏 ≠ 𝑐))
62		eldifsn 4753	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑐 ∈ 𝑉 ∧ 𝑐 ≠ 𝑎))
63		necom 3011	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ≠ 𝑎 ↔ 𝑎 ≠ 𝑐)
64	63	anbi2i 634	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝑉 ∧ 𝑐 ≠ 𝑎) ↔ (𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐))
65	62, 64	bitri 278	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐))
66	65	anbi1i 635	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ (𝑉 ∖ {𝑎}) ∧ 𝑏 ≠ 𝑐) ↔ ((𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐) ∧ 𝑏 ≠ 𝑐))
67		df-3an 1105	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ((𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐) ∧ 𝑏 ≠ 𝑐))
68	66, 67	bitr4i 281	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ (𝑉 ∖ {𝑎}) ∧ 𝑏 ≠ 𝑐) ↔ (𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
69	68	exbii 1878	. . . . . . . . . 10 ⊢ (∃𝑐(𝑐 ∈ (𝑉 ∖ {𝑎}) ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑐(𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
70	61, 69	bitri 278	. . . . . . . . 9 ⊢ (∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 ↔ ∃𝑐(𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
71		df-rex 3090	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑐(𝑐 ∈ 𝑉 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
72	60, 70, 71	3bitr4i 306	. . . . . . . 8 ⊢ (∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 ↔ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
73	72	biimpi 219	. . . . . . 7 ⊢ (∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 → ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
74	58, 73	anim12i 624	. . . . . 6 ⊢ ((𝑏 ∈ (𝑉 ∖ {𝑎}) ∧ ∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐) → ((𝑏 ∈ 𝑉 ∧ ∀𝑐 𝑎 ≠ 𝑏) ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
75		alral 3094	. . . . . . . . . 10 ⊢ (∀𝑐 𝑎 ≠ 𝑏 → ∀𝑐 ∈ 𝑉 𝑎 ≠ 𝑏)
76	75	anim1i 626	. . . . . . . . 9 ⊢ ((∀𝑐 𝑎 ≠ 𝑏 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → (∀𝑐 ∈ 𝑉 𝑎 ≠ 𝑏 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
77		r19.29 3128	. . . . . . . . 9 ⊢ ((∀𝑐 ∈ 𝑉 𝑎 ≠ 𝑏 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
78		3anass 1111	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ (𝑎 ≠ 𝑏 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
79	78	biimpri 231	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 𝑏 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
80	79	reximi 3103	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
81	76, 77, 80	3syl 19	. . . . . . . 8 ⊢ ((∀𝑐 𝑎 ≠ 𝑏 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
82	81	anim2i 628	. . . . . . 7 ⊢ ((𝑏 ∈ 𝑉 ∧ (∀𝑐 𝑎 ≠ 𝑏 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) → (𝑏 ∈ 𝑉 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
83	82	anassrs 472	. . . . . 6 ⊢ (((𝑏 ∈ 𝑉 ∧ ∀𝑐 𝑎 ≠ 𝑏) ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → (𝑏 ∈ 𝑉 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
84	74, 83	syl 18	. . . . 5 ⊢ ((𝑏 ∈ (𝑉 ∖ {𝑎}) ∧ ∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐) → (𝑏 ∈ 𝑉 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
85	84	reximi2 3098	. . . 4 ⊢ (∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
86	85	reximi 3103	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
87	51, 86	sylbir 238	. 2 ⊢ (∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
88	50, 87	syl 18	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 ∀wal 1568 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ∖ cdif 3902 ∪ cun 3903 {csn 4589 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 0cc0 11095 1c1 11096 + caddc 11098 ℝ^*cxr 11237 < clt 11238 2c2 12290 ℕ₀cn0 12499 ♯chash 14362

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-n0 12500 df-xnn0 12573 df-z 12587 df-uz 12858 df-xneg 13132 df-xadd 13133 df-fz 13531 df-hash 14363

This theorem is referenced by: cusgr3cyclex 35628

W3C validator