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Mirrors > Home > MPE Home > Th. List > umgruhgr | Structured version Visualization version GIF version |
Description: An undirected multigraph is an undirected hypergraph. (Contributed by AV, 26-Nov-2020.) |
Ref | Expression |
---|---|
umgruhgr | ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgrupgr 29135 | . 2 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) | |
2 | upgruhgr 29134 | . 2 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 UHGraphcuhgr 29088 UPGraphcupgr 29112 UMGraphcumgr 29113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-i2m1 11221 ax-1ne0 11222 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-2 12327 df-uhgr 29090 df-upgr 29114 df-umgr 29115 |
This theorem is referenced by: umgredgprv 29139 umgr2edg 29241 subumgredg2 29317 subumgr 29320 umgrspan 29326 umgrreslem 29337 umgrres 29339 umgrres1lem 29342 vdumgr0 29513 vtxdumgr0nedg 29526 umgr2wlk 29979 umgrwwlks2on 29987 umgr3cyclex 30212 vdn0conngrumgrv2 30225 umgr2cycllem 35125 isubgrumgr 47795 |
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