| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 30279 | . 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))) |
| 4 | 3 | simplbi 497 | 1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 ∃!wreu 3378 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 {cpr 4628 ‘cfv 6561 Vtxcvtx 29013 Edgcedg 29064 USGraphcusgr 29166 FriendGraph cfrgr 30277 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-frgr 30278 |
| This theorem is referenced by: frgreu 30287 frcond3 30288 nfrgr2v 30291 3vfriswmgr 30297 2pthfrgrrn2 30302 2pthfrgr 30303 3cyclfrgrrn2 30306 3cyclfrgr 30307 n4cyclfrgr 30310 frgrnbnb 30312 vdgn0frgrv2 30314 vdgn1frgrv2 30315 frgrncvvdeqlem2 30319 frgrncvvdeqlem3 30320 frgrncvvdeqlem6 30323 frgrncvvdeqlem9 30326 frgrncvvdeq 30328 frgrwopreglem4a 30329 frgrwopreg 30342 frgrregorufrg 30345 frgr2wwlkeu 30346 frgr2wsp1 30349 frgr2wwlkeqm 30350 frrusgrord0lem 30358 frrusgrord0 30359 friendshipgt3 30417 |
| Copyright terms: Public domain | W3C validator |