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Theorem hashgt23el 14460

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 12367	. . . . . 6 ⊢ 0 < 2
2		0xr 11306	. . . . . . 7 ⊢ 0 ∈ ℝ^*
3		2re 12338	. . . . . . . 8 ⊢ 2 ∈ ℝ
4	3	rexri 11317	. . . . . . 7 ⊢ 2 ∈ ℝ^*
5		hashxrcl 14393	. . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → (♯‘𝑉) ∈ ℝ^*)
6		xrlttr 13179	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 2 ∈ ℝ^* ∧ (♯‘𝑉) ∈ ℝ^*) → ((0 < 2 ∧ 2 < (♯‘𝑉)) → 0 < (♯‘𝑉)))
7	2, 4, 5, 6	mp3an12i 1464	. . . . . 6 ⊢ (𝑉 ∈ 𝑊 → ((0 < 2 ∧ 2 < (♯‘𝑉)) → 0 < (♯‘𝑉)))
8	1, 7	mpani 696	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → (2 < (♯‘𝑉) → 0 < (♯‘𝑉)))
9		hashgt0elex 14437	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 0 < (♯‘𝑉)) → ∃𝑎 𝑎 ∈ 𝑉)
10	9	ex 412	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → (0 < (♯‘𝑉) → ∃𝑎 𝑎 ∈ 𝑉))
11	8, 10	syld 47	. . . 4 ⊢ (𝑉 ∈ 𝑊 → (2 < (♯‘𝑉) → ∃𝑎 𝑎 ∈ 𝑉))
12	11	imp 406	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎 𝑎 ∈ 𝑉)
13		difexg 5335	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → (𝑉 ∖ {𝑎}) ∈ V)
14		difsnid 4815	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑉 → ((𝑉 ∖ {𝑎}) ∪ {𝑎}) = 𝑉)
15	14	fveq2d 6911	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝑉 → (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) = (♯‘𝑉))
16	15	breq2d 5160	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝑉 → (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) ↔ 2 < (♯‘𝑉)))
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊) → (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) ↔ 2 < (♯‘𝑉)))
18		df-2 12327	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
19	18	breq1i 5155	. . . . . . . . . . . 12 ⊢ (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})))
20		neldifsn 4797	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑎 ∈ (𝑉 ∖ {𝑎})
21		1nn0 12540	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
22		hashunsnggt 14430	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑉 ∖ {𝑎}) ∈ V ∧ 𝑎 ∈ 𝑉 ∧ 1 ∈ ℕ₀) ∧ ¬ 𝑎 ∈ (𝑉 ∖ {𝑎})) → (1 < (♯‘(𝑉 ∖ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))))
23	21, 22	mp3anl3 1456	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 ∖ {𝑎}) ∈ V ∧ 𝑎 ∈ 𝑉) ∧ ¬ 𝑎 ∈ (𝑉 ∖ {𝑎})) → (1 < (♯‘(𝑉 ∖ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))))
24	13, 23	sylanl1 680	. . . . . . . . . . . . . 14 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉) ∧ ¬ 𝑎 ∈ (𝑉 ∖ {𝑎})) → (1 < (♯‘(𝑉 ∖ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))))
25	20, 24	mpan2 691	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉) → (1 < (♯‘(𝑉 ∖ {𝑎})) ↔ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))))
26	25	biimp3ar 1469	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉 ∧ (1 + 1) < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))) → 1 < (♯‘(𝑉 ∖ {𝑎})))
27	19, 26	syl3an3b 1404	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉 ∧ 2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎}))) → 1 < (♯‘(𝑉 ∖ {𝑎})))
28	27	3expia 1120	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑉) → (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
29	28	ancoms 458	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊) → (2 < (♯‘((𝑉 ∖ {𝑎}) ∪ {𝑎})) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
30	17, 29	sylbird 260	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊) → (2 < (♯‘𝑉) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
31	30	3impia 1116	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎})))
32	31	3expib 1121	. . . . . 6 ⊢ (𝑎 ∈ 𝑉 → ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
33		1lt2 12435	. . . . . . . . . . 11 ⊢ 1 < 2
34		1xr 11318	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ^*
35		xrlttr 13179	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ^* ∧ 2 ∈ ℝ^* ∧ (♯‘𝑉) ∈ ℝ^*) → ((1 < 2 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘𝑉)))
36	34, 4, 5, 35	mp3an12i 1464	. . . . . . . . . . 11 ⊢ (𝑉 ∈ 𝑊 → ((1 < 2 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘𝑉)))
37	33, 36	mpani 696	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝑊 → (2 < (♯‘𝑉) → 1 < (♯‘𝑉)))
38	37	imp 406	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘𝑉))
39	38	3adant1 1129	. . . . . . . 8 ⊢ ((¬ 𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘𝑉))
40		difsn 4803	. . . . . . . . . 10 ⊢ (¬ 𝑎 ∈ 𝑉 → (𝑉 ∖ {𝑎}) = 𝑉)
41	40	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((¬ 𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → (𝑉 ∖ {𝑎}) = 𝑉)
42	41	fveq2d 6911	. . . . . . . 8 ⊢ ((¬ 𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → (♯‘(𝑉 ∖ {𝑎})) = (♯‘𝑉))
43	39, 42	breqtrrd 5176	. . . . . . 7 ⊢ ((¬ 𝑎 ∈ 𝑉 ∧ 𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎})))
44	43	3expib 1121	. . . . . 6 ⊢ (¬ 𝑎 ∈ 𝑉 → ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎}))))
45	32, 44	pm2.61i 182	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → 1 < (♯‘(𝑉 ∖ {𝑎})))
46		hashgt12el 14458	. . . . 5 ⊢ (((𝑉 ∖ {𝑎}) ∈ V ∧ 1 < (♯‘(𝑉 ∖ {𝑎}))) → ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐)
47	13, 45, 46	syl2an2r 685	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐)
48	47	alrimiv 1925	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∀𝑎∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐)
49		19.29r 1872	. . 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 584	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐))
51		df-rex 3069	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 ↔ ∃𝑎(𝑎 ∈ 𝑉 ∧ ∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐))
52		eldifsn 4791	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑏 ∈ 𝑉 ∧ 𝑏 ≠ 𝑎))
53		necom 2992	. . . . . . . . . 10 ⊢ (𝑏 ≠ 𝑎 ↔ 𝑎 ≠ 𝑏)
54	53	anbi2i 623	. . . . . . . . 9 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑏 ≠ 𝑎) ↔ (𝑏 ∈ 𝑉 ∧ 𝑎 ≠ 𝑏))
55	52, 54	bitri 275	. . . . . . . 8 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑏 ∈ 𝑉 ∧ 𝑎 ≠ 𝑏))
56		ax-5 1908	. . . . . . . . 9 ⊢ (𝑎 ≠ 𝑏 → ∀𝑐 𝑎 ≠ 𝑏)
57	56	anim2i 617	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑎 ≠ 𝑏) → (𝑏 ∈ 𝑉 ∧ ∀𝑐 𝑎 ≠ 𝑏))
58	55, 57	sylbi 217	. . . . . . 7 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → (𝑏 ∈ 𝑉 ∧ ∀𝑐 𝑎 ≠ 𝑏))
59		3anass 1094	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ (𝑐 ∈ 𝑉 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
60	59	exbii 1845	. . . . . . . . 9 ⊢ (∃𝑐(𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑐(𝑐 ∈ 𝑉 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
61		df-rex 3069	. . . . . . . . . 10 ⊢ (∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 ↔ ∃𝑐(𝑐 ∈ (𝑉 ∖ {𝑎}) ∧ 𝑏 ≠ 𝑐))
62		eldifsn 4791	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑐 ∈ 𝑉 ∧ 𝑐 ≠ 𝑎))
63		necom 2992	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ≠ 𝑎 ↔ 𝑎 ≠ 𝑐)
64	63	anbi2i 623	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝑉 ∧ 𝑐 ≠ 𝑎) ↔ (𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐))
65	62, 64	bitri 275	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝑉 ∖ {𝑎}) ↔ (𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐))
66	65	anbi1i 624	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ (𝑉 ∖ {𝑎}) ∧ 𝑏 ≠ 𝑐) ↔ ((𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐) ∧ 𝑏 ≠ 𝑐))
67		df-3an 1088	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ((𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐) ∧ 𝑏 ≠ 𝑐))
68	66, 67	bitr4i 278	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ (𝑉 ∖ {𝑎}) ∧ 𝑏 ≠ 𝑐) ↔ (𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
69	68	exbii 1845	. . . . . . . . . 10 ⊢ (∃𝑐(𝑐 ∈ (𝑉 ∖ {𝑎}) ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑐(𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
70	61, 69	bitri 275	. . . . . . . . 9 ⊢ (∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 ↔ ∃𝑐(𝑐 ∈ 𝑉 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
71		df-rex 3069	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑐(𝑐 ∈ 𝑉 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
72	60, 70, 71	3bitr4i 303	. . . . . . . 8 ⊢ (∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 ↔ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
73	72	biimpi 216	. . . . . . 7 ⊢ (∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 → ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
74	58, 73	anim12i 613	. . . . . 6 ⊢ ((𝑏 ∈ (𝑉 ∖ {𝑎}) ∧ ∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐) → ((𝑏 ∈ 𝑉 ∧ ∀𝑐 𝑎 ≠ 𝑏) ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
75		alral 3073	. . . . . . . . . 10 ⊢ (∀𝑐 𝑎 ≠ 𝑏 → ∀𝑐 ∈ 𝑉 𝑎 ≠ 𝑏)
76	75	anim1i 615	. . . . . . . . 9 ⊢ ((∀𝑐 𝑎 ≠ 𝑏 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → (∀𝑐 ∈ 𝑉 𝑎 ≠ 𝑏 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
77		r19.29 3112	. . . . . . . . 9 ⊢ ((∀𝑐 ∈ 𝑉 𝑎 ≠ 𝑏 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
78		3anass 1094	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ (𝑎 ≠ 𝑏 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
79	78	biimpri 228	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 𝑏 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
80	79	reximi 3082	. . . . . . . . 9 ⊢ (∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
81	76, 77, 80	3syl 18	. . . . . . . 8 ⊢ ((∀𝑐 𝑎 ≠ 𝑏 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) → ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
82	81	anim2i 617	. . . . . . 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 467	. . . . . 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 17	. . . . 5 ⊢ ((𝑏 ∈ (𝑉 ∖ {𝑎}) ∧ ∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐) → (𝑏 ∈ 𝑉 ∧ ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))
85	84	reximi2 3077	. . . 4 ⊢ (∃𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑐 ∈ (𝑉 ∖ {𝑎})𝑏 ≠ 𝑐 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
86	85	reximi 3082	. . 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 235	. 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 17	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 206 ∧ wa 395 ∧ w3a 1086 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 ℝ^*cxr 11292 < clt 11293 2c2 12319 ℕ₀cn0 12524 ♯chash 14366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-xneg 13152 df-xadd 13153 df-fz 13545 df-hash 14367

This theorem is referenced by: cusgr3cyclex 35121

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