![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 28194 | . 2 ⊢ ∅ ∈ USGraph | |
2 | ral0 4471 | . 2 ⊢ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅) | |
3 | vtxval0 27993 | . . . 4 ⊢ (Vtx‘∅) = ∅ | |
4 | 3 | eqcomi 2746 | . . 3 ⊢ ∅ = (Vtx‘∅) |
5 | eqid 2737 | . . 3 ⊢ (Edg‘∅) = (Edg‘∅) | |
6 | 4, 5 | isfrgr 29207 | . 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 2107 ∀wral 3065 ∃!wreu 3352 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4283 {csn 4587 {cpr 4589 ‘cfv 6497 Vtxcvtx 27950 Edgcedg 28001 USGraphcusgr 28103 FriendGraph cfrgr 29205 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-ltxr 11195 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-dec 12620 df-slot 17055 df-ndx 17067 df-base 17085 df-edgf 27941 df-vtx 27952 df-iedg 27953 df-usgr 28105 df-frgr 29206 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |