grtriprop - Mathbox for Alexander van der Vekens

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Theorem grtriprop 47846

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 6945	. . . . . 6 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → 𝐺 ∈ V)
2		grtri.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
3		grtri.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	grtri 47845	. . . . . 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 2825	. . . 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 6839	. . . . . . 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 1919	. . . . 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 3695	. . . 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 7466	. . . . . . . 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 14389	. . . . . . 7 ⊢ ((𝑇 ∈ 𝒫 𝑉 ∧ 𝑓:(0..^3)–1-1-onto→𝑇) → (♯‘(0..^3)) = (♯‘𝑇))
15		eqcom 2742	. . . . . . . . 9 ⊢ ((♯‘(0..^3)) = (♯‘𝑇) ↔ (♯‘𝑇) = (♯‘(0..^3)))
16		3nn0 12542	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
17		hashfzo0 14466	. . . . . . . . . . 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 2746	. . . . . . . . 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 14531	. . . . . . . . . . . 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 4612	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ 𝒫 𝑉 → 𝑇 ⊆ 𝑉)
25		ss2rexv 4067	. . . . . . . . . . . . . . 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 4065	. . . . . . . . . . . . . . . . 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 3168	. . . . . . . . . . . . . . . 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 3168	. . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (𝑓:(0..^3)–1-1-onto→𝑇 ↔ 𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧}))
36		grtriproplem 47844	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . 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 3166	. . . . . . . . . . . . . 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 3205	. . . . . . . . . . . . 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 1931	. . . 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 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 {cpr 4633 {ctp 4635 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 2c2 12319 3c3 12320 ℕ₀cn0 12524 ..^cfzo 13691 ♯chash 14366 Vtxcvtx 29028 Edgcedg 29079 GrTrianglescgrtri 47842

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-rep 5285 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-tp 4636 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-2o 8506 df-3o 8507 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-3 12328 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-grtri 47843

This theorem is referenced by: grtrif1o 47847 isgrtri 47848 grtrissvtx 47849 grimgrtri 47852 grlimgrtri 47899

W3C validator