grtriprop - Mathbox for Alexander van der Vekens

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Theorem grtriprop 48129

Description: The properties of a triangle. (Contributed by AV, 25-Jul-2025.)

**Hypotheses**
Ref	Expression
grtri.v	⊢ 𝑉 = (Vtx‘𝐺)
grtri.e	⊢ 𝐸 = (Edg‘𝐺)

**Assertion**
Ref	Expression
grtriprop	⊢ (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))

Distinct variable groups: 𝑥,𝐸,𝑦,𝑧 𝑥,𝑇,𝑦,𝑧 𝑥,𝑉,𝑦,𝑧 Allowed substitution hints: 𝐺(𝑥,𝑦,𝑧)

Proof of Theorem grtriprop Dummy variables 𝑓 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6867	. . . . . 6 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → 𝐺 ∈ V)
2		grtri.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
3		grtri.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	grtri 48128	. . . . . 6 ⊢ (𝐺 ∈ V → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
5	1, 4	syl 17	. . . . 5 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
6	5	eleq2d 2820	. . . 4 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → (𝑇 ∈ (GrTriangles‘𝐺) ↔ 𝑇 ∈ {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))}))
7		f1oeq3 6762	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑓:(0..^3)–1-1-onto→𝑡 ↔ 𝑓:(0..^3)–1-1-onto→𝑇))
8	7	anbi1d 631	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ↔ (𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
9	8	exbidv 1922	. . . . 5 ⊢ (𝑡 = 𝑇 → (∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ↔ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
10	9	elrab 3644	. . . 4 ⊢ (𝑇 ∈ {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))} ↔ (𝑇 ∈ 𝒫 𝑉 ∧ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
11	6, 10	bitrdi 287	. . 3 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → (𝑇 ∈ (GrTriangles‘𝐺) ↔ (𝑇 ∈ 𝒫 𝑉 ∧ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)))))
12		ovexd 7391	. . . . . . . 8 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → (0..^3) ∈ V)
13		simpr 484	. . . . . . . 8 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → 𝑓:(0..^3)–1-1-onto→𝑇)
14	12, 13	hasheqf1od 14274	. . . . . . 7 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → (♯‘(0..^3)) = (♯‘𝑇))
15		eqcom 2741	. . . . . . . . 9 ⊢ ((♯‘(0..^3)) = (♯‘𝑇) ↔ (♯‘𝑇) = (♯‘(0..^3)))
16		3nn0 12417	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
17		hashfzo0 14351	. . . . . . . . . . 11 ⊢ (3 ∈ ℕ₀ → (♯‘(0..^3)) = 3)
18	16, 17	mp1i 13	. . . . . . . . . 10 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → (♯‘(0..^3)) = 3)
19	18	eqeq2d 2745	. . . . . . . . 9 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → ((♯‘𝑇) = (♯‘(0..^3)) ↔ (♯‘𝑇) = 3))
20	15, 19	bitrid 283	. . . . . . . 8 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → ((♯‘(0..^3)) = (♯‘𝑇) ↔ (♯‘𝑇) = 3))
21		hash3tpb 14416	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ 𝒫 𝑉 → ((♯‘𝑇) = 3 ↔ ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
22	21	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → ((♯‘𝑇) = 3 ↔ ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
23	22	biimpa 476	. . . . . . . . . 10 ⊢ (((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) → ∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}))
24		elpwi 4559	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ 𝒫 𝑉 → 𝑇 ⊆ 𝑉)
25		ss2rexv 4003	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ⊆ 𝑉 → (∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
26		ssrexv 4001	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ⊆ 𝑉 → (∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
27	26	reximdv 3149	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ⊆ 𝑉 → (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
28	27	reximdv 3149	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ⊆ 𝑉 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
29	25, 28	syld 47	. . . . . . . . . . . . . 14 ⊢ (𝑇 ⊆ 𝑉 → (∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
30	24, 29	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ 𝒫 𝑉 → (∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
31	30	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → (∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
32	31	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) → (∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
33		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑧 ∈ 𝑉) ∧ ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})) → 𝑇 = {𝑥, 𝑦, 𝑧})
34		simp-5r 785	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑧 ∈ 𝑉) ∧ ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})) → (♯‘𝑇) = 3)
35		f1oeq3 6762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (𝑓:(0..^3)–1-1-onto→𝑇 ↔ 𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧}))
36		grtriproplem 48127	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))
37	36	2a1d 26	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑧 ∈ 𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
38	37	ex 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑧 ∈ 𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
39	38	a1d 25	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} → ((♯‘𝑇) = 3 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑧 ∈ 𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))))
40	35, 39	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (𝑓:(0..^3)–1-1-onto→𝑇 → ((♯‘𝑇) = 3 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑧 ∈ 𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))))
41	40	adantld 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → ((♯‘𝑇) = 3 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑧 ∈ 𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))))
42	41	imp4c 423	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → ((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑧 ∈ 𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
43	42	imp4c 423	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → ((((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑧 ∈ 𝑉) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
44	43	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ((((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑧 ∈ 𝑉) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
45	44	impcom 407	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑧 ∈ 𝑉) ∧ ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))
46	33, 34, 45	3jca 1128	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑧 ∈ 𝑉) ∧ ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})) → (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
47	46	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ 𝑧 ∈ 𝑉) → (((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
48	47	reximdva 3147	. . . . . . . . . . . . . 14 ⊢ (((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
49	48	reximdvva 3182	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
50	49	ex 412	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
51	50	com23 86	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
52	32, 51	syld 47	. . . . . . . . . 10 ⊢ (((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) → (∃𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑇 ((𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
53	23, 52	mpd 15	. . . . . . . . 9 ⊢ (((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) ∧ (♯‘𝑇) = 3) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
54	53	ex 412	. . . . . . . 8 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → ((♯‘𝑇) = 3 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
55	20, 54	sylbid 240	. . . . . . 7 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → ((♯‘(0..^3)) = (♯‘𝑇) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
56	14, 55	mpd 15	. . . . . 6 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
57	56	expimpd 453	. . . . 5 ⊢ (𝑇 ∈ 𝒫 𝑉 → ((𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
58	57	exlimdv 1934	. . . 4 ⊢ (𝑇 ∈ 𝒫 𝑉 → (∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
59	58	imp 406	. . 3 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
60	11, 59	biimtrdi 253	. 2 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
61	60	pm2.43i 52	1 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ≠ wne 2930 ∃wrex 3058 {crab 3397 Vcvv 3438 ⊆ wss 3899 𝒫 cpw 4552 {cpr 4580 {ctp 4582 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7356 0cc0 11024 1c1 11025 2c2 12198 3c3 12199 ℕ₀cn0 12399 ..^cfzo 13568 ♯chash 14251 Vtxcvtx 29018 Edgcedg 29069 GrTrianglescgrtri 48125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-3o 8397 df-oadd 8399 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-xnn0 12473 df-z 12487 df-uz 12750 df-fz 13422 df-fzo 13569 df-hash 14252 df-grtri 48126

This theorem is referenced by: grtrif1o 48130 isgrtri 48131 grtrissvtx 48132 grimgrtri 48137 grlimgrtri 48191

W3C validator