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Mirrors > Home > MPE Home > Th. List > frcond2 | Structured version Visualization version GIF version |
Description: The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.) |
Ref | Expression |
---|---|
frcond1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
frcond1.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
frcond2 | ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frcond1.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | frcond1.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | frcond1 30023 | . 2 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)) |
4 | prex 5425 | . . . . 5 ⊢ {𝐴, 𝑏} ∈ V | |
5 | prex 5425 | . . . . 5 ⊢ {𝑏, 𝐶} ∈ V | |
6 | 4, 5 | prss 4818 | . . . 4 ⊢ (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸) |
7 | 6 | bicomi 223 | . . 3 ⊢ ({{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)) |
8 | 7 | reubii 3379 | . 2 ⊢ (∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)) |
9 | 3, 8 | imbitrdi 250 | 1 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∃!wreu 3368 ⊆ wss 3943 {cpr 4625 ‘cfv 6536 Vtxcvtx 28759 Edgcedg 28810 FriendGraph cfrgr 30015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6488 df-fv 6544 df-frgr 30016 |
This theorem is referenced by: frgreu 30025 frgrncvvdeqlem9 30064 frgr2wwlkeu 30084 numclwwlk2lem1 30133 |
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