grtriprop - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > grtriprop		Structured version Visualization version GIF version

Theorem grtriprop 47953

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 6914	. . . . . 6 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → 𝐺 ∈ V)
2		grtri.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
3		grtri.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	grtri 47952	. . . . . 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 6808	. . . . . . 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 1921	. . . . 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 3671	. . . 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 7440	. . . . . . . 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 14371	. . . . . . 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 12519	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
17		hashfzo0 14448	. . . . . . . . . . 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 14513	. . . . . . . . . . . 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 4582	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ 𝒫 𝑉 → 𝑇 ⊆ 𝑉)
25		ss2rexv 4030	. . . . . . . . . . . . . . 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 4028	. . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . 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 6808	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (𝑓:(0..^3)–1-1-onto→𝑇 ↔ 𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧}))
36		grtriproplem 47951	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . . 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 3192	. . . . . . . . . . . . 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 1933	. . . 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 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2932 ∃wrex 3060 {crab 3415 Vcvv 3459 ⊆ wss 3926 𝒫 cpw 4575 {cpr 4603 {ctp 4605 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7405 0cc0 11129 1c1 11130 2c2 12295 3c3 12296 ℕ₀cn0 12501 ..^cfzo 13671 ♯chash 14348 Vtxcvtx 28975 Edgcedg 29026 GrTrianglescgrtri 47949

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-3o 8482 df-oadd 8484 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-hash 14349 df-grtri 47950

This theorem is referenced by: grtrif1o 47954 isgrtri 47955 grtrissvtx 47956 grimgrtri 47961 grlimgrtri 48008

W3C validator