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Mirrors > Home > MPE Home > Th. List > frgraun | Structured version Visualization version GIF version |
Description: Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) |
Ref | Expression |
---|---|
frgraun | ⊢ (𝑉 FriendGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgraunss 26522 | . 2 ⊢ (𝑉 FriendGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸)) | |
2 | prex 4836 | . . . . 5 ⊢ {𝐴, 𝑏} ∈ V | |
3 | prex 4836 | . . . . 5 ⊢ {𝑏, 𝐶} ∈ V | |
4 | 2, 3 | prss 4291 | . . . 4 ⊢ (({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸) |
5 | 4 | bicomi 213 | . . 3 ⊢ ({{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)) |
6 | 5 | reubii 3105 | . 2 ⊢ (∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸 ↔ ∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)) |
7 | 1, 6 | syl6ib 240 | 1 ⊢ (𝑉 FriendGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ≠ wne 2780 ∃!wreu 2898 ⊆ wss 3540 {cpr 4127 class class class wbr 4583 ran crn 5039 FriendGrph cfrgra 26515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-frgra 26516 |
This theorem is referenced by: frgrancvvdeqlemC 26566 frgraeu 26581 frg2woteu 26582 numclwwlk2lem1 26629 |
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