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Mirrors > Home > MPE Home > Th. List > Mathboxes > clnbgrssedg | Structured version Visualization version GIF version |
Description: The vertices connected by an edge are a subset of the neighborhood of each of these vertices. (Contributed by AV, 26-May-2025.) (Proof shortened by AV, 24-Aug-2025.) |
Ref | Expression |
---|---|
clnbgrssedg.e | ⊢ 𝐸 = (Edg‘𝐺) |
clnbgrssedg.n | ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋) |
Ref | Expression |
---|---|
clnbgrssedg | ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clnbgrssedg.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
2 | clnbgrssedg.n | . . . . 5 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋) | |
3 | 1, 2 | clnbgredg 47713 | . . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾)) → 𝑣 ∈ 𝑁) |
4 | 3 | 3exp2 1352 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ 𝐸 → (𝑋 ∈ 𝐾 → (𝑣 ∈ 𝐾 → 𝑣 ∈ 𝑁)))) |
5 | 4 | 3imp 1109 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → (𝑣 ∈ 𝐾 → 𝑣 ∈ 𝑁)) |
6 | 5 | ssrdv 4001 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1535 ∈ wcel 2104 ⊆ wss 3963 ‘cfv 6558 (class class class)co 7425 Edgcedg 29060 UHGraphcuhgr 29069 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-fn 6561 df-f 6562 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-1st 8007 df-2nd 8008 df-edg 29061 df-uhgr 29071 df-clnbgr 47694 |
This theorem is referenced by: grlimgrtrilem1 47819 |
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