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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 2818 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2818 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | isfrgr 27966 | . 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))) |
4 | 3 | simplbi 498 | 1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∀wral 3135 ∃!wreu 3137 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 {cpr 4559 ‘cfv 6348 Vtxcvtx 26708 Edgcedg 26759 USGraphcusgr 26861 FriendGraph cfrgr 27964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-frgr 27965 |
This theorem is referenced by: frgreu 27974 frcond3 27975 nfrgr2v 27978 3vfriswmgr 27984 2pthfrgrrn2 27989 2pthfrgr 27990 3cyclfrgrrn2 27993 3cyclfrgr 27994 n4cyclfrgr 27997 frgrnbnb 27999 vdgn0frgrv2 28001 vdgn1frgrv2 28002 frgrncvvdeqlem2 28006 frgrncvvdeqlem3 28007 frgrncvvdeqlem6 28010 frgrncvvdeqlem9 28013 frgrncvvdeq 28015 frgrwopreglem4a 28016 frgrwopreg 28029 frgrregorufrg 28032 frgr2wwlkeu 28033 frgr2wsp1 28036 frgr2wwlkeqm 28037 frrusgrord0lem 28045 frrusgrord0 28046 friendshipgt3 28104 |
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