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Mirrors > Home > MPE Home > Th. List > frgrncvvdeqlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for frgrncvvdeq 28080. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.) |
Ref | Expression |
---|---|
frgrncvvdeq.v1 | ⊢ 𝑉 = (Vtx‘𝐺) |
frgrncvvdeq.e | ⊢ 𝐸 = (Edg‘𝐺) |
frgrncvvdeq.nx | ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) |
frgrncvvdeq.ny | ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) |
frgrncvvdeq.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
frgrncvvdeq.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
frgrncvvdeq.ne | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
frgrncvvdeq.xy | ⊢ (𝜑 → 𝑌 ∉ 𝐷) |
frgrncvvdeq.f | ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) |
frgrncvvdeq.a | ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) |
Ref | Expression |
---|---|
frgrncvvdeqlem1 | ⊢ (𝜑 → 𝑋 ∉ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrncvvdeq.xy | . . . 4 ⊢ (𝜑 → 𝑌 ∉ 𝐷) | |
2 | df-nel 3122 | . . . . 5 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ 𝐷) | |
3 | frgrncvvdeq.nx | . . . . . 6 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) | |
4 | 3 | eleq2i 2902 | . . . . 5 ⊢ (𝑌 ∈ 𝐷 ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) |
5 | 2, 4 | xchbinx 336 | . . . 4 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) |
6 | 1, 5 | sylib 220 | . . 3 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) |
7 | nbgrsym 27137 | . . 3 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) | |
8 | 6, 7 | sylnibr 331 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌)) |
9 | frgrncvvdeq.ny | . . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) | |
10 | neleq2 3127 | . . . 4 ⊢ (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌))) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ (𝑋 ∉ 𝑁 ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑌)) |
12 | df-nel 3122 | . . 3 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌)) | |
13 | 11, 12 | bitri 277 | . 2 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌)) |
14 | 8, 13 | sylibr 236 | 1 ⊢ (𝜑 → 𝑋 ∉ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∉ wnel 3121 {cpr 4561 ↦ cmpt 5137 ‘cfv 6348 ℩crio 7105 (class class class)co 7148 Vtxcvtx 26773 Edgcedg 26824 NeighbVtx cnbgr 27106 FriendGraph cfrgr 28029 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-nbgr 27107 |
This theorem is referenced by: frgrncvvdeqlem7 28076 frgrncvvdeqlem8 28077 frgrncvvdeqlem9 28078 |
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