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Mirrors > Home > MPE Home > Th. List > Mathboxes > predgclnbgrel | Structured version Visualization version GIF version |
Description: If a (not necessarily proper) unordered pair containing a vertex is an edge, the other vertex is in the closed neighborhood of the first vertex. (Contributed by AV, 23-Aug-2025.) |
Ref | Expression |
---|---|
predgclnbgrel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
predgclnbgrel.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
predgclnbgrel | ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 1146 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) | |
2 | simp3 1136 | . . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ∈ 𝐸) | |
3 | sseq2 4022 | . . . . 5 ⊢ (𝑒 = {𝑋, 𝑁} → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁})) | |
4 | 3 | adantl 481 | . . . 4 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) ∧ 𝑒 = {𝑋, 𝑁}) → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁})) |
5 | ssidd 4019 | . . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ⊆ {𝑋, 𝑁}) | |
6 | 2, 4, 5 | rspcedvd 3624 | . . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) |
7 | 6 | olcd 873 | . 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) |
8 | predgclnbgrel.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
9 | predgclnbgrel.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
10 | 8, 9 | clnbgrel 47702 | . 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))) |
11 | 1, 7, 10 | sylanbrc 582 | 1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1535 ∈ wcel 2104 ∃wrex 3066 ⊆ wss 3963 {cpr 4632 ‘cfv 6558 (class class class)co 7425 Vtxcvtx 29009 Edgcedg 29060 ClNeighbVtx cclnbgr 47693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-1st 8007 df-2nd 8008 df-clnbgr 47694 |
This theorem is referenced by: grlimgrtri 47821 |
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