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Mirrors > Home > MPE Home > Th. List > cusgr0 | Structured version Visualization version GIF version |
Description: The null graph (with no vertices and no edges) represented by the empty set is a complete simple graph. (Contributed by AV, 1-Nov-2020.) |
Ref | Expression |
---|---|
cusgr0 | ⊢ ∅ ∈ ComplUSGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr0 26952 | . 2 ⊢ ∅ ∈ USGraph | |
2 | cplgr0 27134 | . 2 ⊢ ∅ ∈ ComplGraph | |
3 | iscusgr 27127 | . 2 ⊢ (∅ ∈ ComplUSGraph ↔ (∅ ∈ USGraph ∧ ∅ ∈ ComplGraph)) | |
4 | 1, 2, 3 | mpbir2an 707 | 1 ⊢ ∅ ∈ ComplUSGraph |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∅c0 4288 USGraphcusgr 26861 ComplGraphccplgr 27118 ComplUSGraphccusgr 27119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fv 6356 df-ov 7148 df-slot 16475 df-base 16477 df-edgf 26702 df-vtx 26710 df-iedg 26711 df-usgr 26863 df-uvtx 27095 df-cplgr 27120 df-cusgr 27121 |
This theorem is referenced by: (None) |
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