Theorem 1vwmgr 28640

Description: Every graph with one vertex (which may be connect with itself by (multiple) loops!) is a windmill graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Revised by AV, 31-Mar-2021.)

Assertion

Ref	Expression
1vwmgr	⊢ ((𝐴 ∈ 𝑋 ∧ 𝑉 = {𝐴}) → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))

Distinct variable groups:   𝐴,ℎ,𝑣,𝑤   ℎ,𝐸   ℎ,𝑉,𝑣,𝑤

Allowed substitution hints:   𝐸(𝑤,𝑣)   𝑋(𝑤,𝑣,ℎ)

Proof of Theorem 1vwmgr

Step	Hyp	Ref	Expression
1		ral0 4443	. . . 4 ⊢ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)
2		sneq 4571	. . . . . . . 8 ⊢ (ℎ = 𝐴 → {ℎ} = {𝐴})
3	2	difeq2d 4057	. . . . . . 7 ⊢ (ℎ = 𝐴 → ({𝐴} ∖ {ℎ}) = ({𝐴} ∖ {𝐴}))
4		difid 4304	. . . . . . 7 ⊢ ({𝐴} ∖ {𝐴}) = ∅
5	3, 4	eqtrdi 2794	. . . . . 6 ⊢ (ℎ = 𝐴 → ({𝐴} ∖ {ℎ}) = ∅)
6		preq2 4670	. . . . . . . 8 ⊢ (ℎ = 𝐴 → {𝑣, ℎ} = {𝑣, 𝐴})
7	6	eleq1d 2823	. . . . . . 7 ⊢ (ℎ = 𝐴 → ({𝑣, ℎ} ∈ 𝐸 ↔ {𝑣, 𝐴} ∈ 𝐸))
8		reueq1 3344	. . . . . . . 8 ⊢ (({𝐴} ∖ {ℎ}) = ({𝐴} ∖ {𝐴}) → (∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
9	3, 8	syl 17	. . . . . . 7 ⊢ (ℎ = 𝐴 → (∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸))
10	7, 9	anbi12d 631	. . . . . 6 ⊢ (ℎ = 𝐴 → (({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
11	5, 10	raleqbidv 3336	. . . . 5 ⊢ (ℎ = 𝐴 → (∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
12	11	rexsng 4610	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ∅ ({𝑣, 𝐴} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {𝐴}){𝑣, 𝑤} ∈ 𝐸)))
13	1, 12	mpbiri 257	. . 3 ⊢ (𝐴 ∈ 𝑋 → ∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
14	13	adantr 481	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑉 = {𝐴}) → ∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
15		difeq1 4050	. . . . 5 ⊢ (𝑉 = {𝐴} → (𝑉 ∖ {ℎ}) = ({𝐴} ∖ {ℎ}))
16		reueq1 3344	. . . . . . 7 ⊢ ((𝑉 ∖ {ℎ}) = ({𝐴} ∖ {ℎ}) → (∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑉 = {𝐴} → (∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))
18	17	anbi2d 629	. . . . 5 ⊢ (𝑉 = {𝐴} → (({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
19	15, 18	raleqbidv 3336	. . . 4 ⊢ (𝑉 = {𝐴} → (∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
20	19	rexeqbi1dv 3341	. . 3 ⊢ (𝑉 = {𝐴} → (∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
21	20	adantl 482	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑉 = {𝐴}) → (∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸) ↔ ∃ℎ ∈ {𝐴}∀𝑣 ∈ ({𝐴} ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ ({𝐴} ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)))
22	14, 21	mpbird 256	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑉 = {𝐴}) → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066   ∖ cdif 3884  ∅c0 4256  {csn 4561  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564

This theorem is referenced by:  1to2vfriswmgr  28643