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Mirrors > Home > MPE Home > Th. List > frgr0 | Structured version Visualization version GIF version |
Description: The null graph (graph without vertices) is a friendship graph. (Contributed by AV, 29-Mar-2021.) |
Ref | Expression |
---|---|
frgr0 | ⊢ ∅ ∈ FriendGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr0 27033 | . 2 ⊢ ∅ ∈ USGraph | |
2 | ral0 4414 | . 2 ⊢ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅) | |
3 | vtxval0 26832 | . . . 4 ⊢ (Vtx‘∅) = ∅ | |
4 | 3 | eqcomi 2807 | . . 3 ⊢ ∅ = (Vtx‘∅) |
5 | eqid 2798 | . . 3 ⊢ (Edg‘∅) = (Edg‘∅) | |
6 | 4, 5 | isfrgr 28045 | . 2 ⊢ (∅ ∈ FriendGraph ↔ (∅ ∈ USGraph ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅))) |
7 | 1, 2, 6 | mpbir2an 710 | 1 ⊢ ∅ ∈ FriendGraph |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ∀wral 3106 ∃!wreu 3108 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 {csn 4525 {cpr 4527 ‘cfv 6324 Vtxcvtx 26789 Edgcedg 26840 USGraphcusgr 26942 FriendGraph cfrgr 28043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fv 6332 df-slot 16479 df-base 16481 df-edgf 26783 df-vtx 26791 df-iedg 26792 df-usgr 26944 df-frgr 28044 |
This theorem is referenced by: (None) |
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