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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 2738 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2738 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | isfrgr 28624 | . 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 2106 ∀wral 3064 ∃!wreu 3066 ∖ cdif 3884 ⊆ wss 3887 {csn 4561 {cpr 4563 ‘cfv 6433 Vtxcvtx 27366 Edgcedg 27417 USGraphcusgr 27519 FriendGraph cfrgr 28622 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-frgr 28623 |
This theorem is referenced by: frgreu 28632 frcond3 28633 nfrgr2v 28636 3vfriswmgr 28642 2pthfrgrrn2 28647 2pthfrgr 28648 3cyclfrgrrn2 28651 3cyclfrgr 28652 n4cyclfrgr 28655 frgrnbnb 28657 vdgn0frgrv2 28659 vdgn1frgrv2 28660 frgrncvvdeqlem2 28664 frgrncvvdeqlem3 28665 frgrncvvdeqlem6 28668 frgrncvvdeqlem9 28671 frgrncvvdeq 28673 frgrwopreglem4a 28674 frgrwopreg 28687 frgrregorufrg 28690 frgr2wwlkeu 28691 frgr2wsp1 28694 frgr2wwlkeqm 28695 frrusgrord0lem 28703 frrusgrord0 28704 friendshipgt3 28762 |
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