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Mirrors > Home > MPE Home > Th. List > frgrusgr | Structured version Visualization version GIF version |
Description: A friendship graph is a simple graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
Ref | Expression |
---|---|
frgrusgr | ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2737 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | isfrgr 29207 | . 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))) |
4 | 3 | simplbi 499 | 1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3065 ∃!wreu 3352 ∖ cdif 3908 ⊆ wss 3911 {csn 4587 {cpr 4589 ‘cfv 6497 Vtxcvtx 27950 Edgcedg 28001 USGraphcusgr 28103 FriendGraph cfrgr 29205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-nul 5264 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-frgr 29206 |
This theorem is referenced by: frgreu 29215 frcond3 29216 nfrgr2v 29219 3vfriswmgr 29225 2pthfrgrrn2 29230 2pthfrgr 29231 3cyclfrgrrn2 29234 3cyclfrgr 29235 n4cyclfrgr 29238 frgrnbnb 29240 vdgn0frgrv2 29242 vdgn1frgrv2 29243 frgrncvvdeqlem2 29247 frgrncvvdeqlem3 29248 frgrncvvdeqlem6 29251 frgrncvvdeqlem9 29254 frgrncvvdeq 29256 frgrwopreglem4a 29257 frgrwopreg 29270 frgrregorufrg 29273 frgr2wwlkeu 29274 frgr2wsp1 29277 frgr2wwlkeqm 29278 frrusgrord0lem 29286 frrusgrord0 29287 friendshipgt3 29345 |
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