Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > frgrusgrfrcond | Structured version Visualization version GIF version |
Description: A friendship graph is a simple graph which fulfils the friendship condition. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
Ref | Expression |
---|---|
isfrgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isfrgr.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
frgrusgrfrcond | ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfrgr.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | isfrgr.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | isfrgr 41430 | . . . 4 ⊢ (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))) |
4 | simpl 472 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸) → 𝐺 ∈ USGraph ) | |
5 | 3, 4 | syl6bi 242 | . . 3 ⊢ (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )) |
6 | 5 | pm2.43i 50 | . 2 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph ) |
7 | 1, 2 | isfrgr 41430 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))) |
8 | 6, 4, 7 | pm5.21nii 367 | 1 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃!wreu 2898 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 {cpr 4127 ‘cfv 5804 Vtxcvtx 25673 Edgcedga 25792 USGraph cusgr 40379 FriendGraph cfrgr 41428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-frgr 41429 |
This theorem is referenced by: frgrusgr 41432 frgr0v 41433 frgr0 41436 frcond1 41438 frcond3 41440 frgr1v 41441 nfrgr2v 41442 frgr3v 41445 2pthfrgrrn 41452 n4cyclfrgr 41461 |
Copyright terms: Public domain | W3C validator |