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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 2730 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2730 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | isfrgr 29780 | . 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))) |
4 | 3 | simplbi 496 | 1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2104 ∀wral 3059 ∃!wreu 3372 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 {cpr 4629 ‘cfv 6542 Vtxcvtx 28523 Edgcedg 28574 USGraphcusgr 28676 FriendGraph cfrgr 29778 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 df-frgr 29779 |
This theorem is referenced by: frgreu 29788 frcond3 29789 nfrgr2v 29792 3vfriswmgr 29798 2pthfrgrrn2 29803 2pthfrgr 29804 3cyclfrgrrn2 29807 3cyclfrgr 29808 n4cyclfrgr 29811 frgrnbnb 29813 vdgn0frgrv2 29815 vdgn1frgrv2 29816 frgrncvvdeqlem2 29820 frgrncvvdeqlem3 29821 frgrncvvdeqlem6 29824 frgrncvvdeqlem9 29827 frgrncvvdeq 29829 frgrwopreglem4a 29830 frgrwopreg 29843 frgrregorufrg 29846 frgr2wwlkeu 29847 frgr2wsp1 29850 frgr2wwlkeqm 29851 frrusgrord0lem 29859 frrusgrord0 29860 friendshipgt3 29918 |
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