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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 26591 | . 2 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) | |
2 | upgruhgr 26590 | . 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 2050 UHGraphcuhgr 26544 UPGraphcupgr 26568 UMGraphcumgr 26569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-i2m1 10403 ax-1ne0 10404 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-2 11503 df-uhgr 26546 df-upgr 26570 df-umgr 26571 |
This theorem is referenced by: umgredgprv 26595 umgr2edg 26694 subumgredg2 26770 subumgr 26773 umgrspan 26779 umgrreslem 26790 umgrres 26792 umgrres1lem 26795 vdumgr0 26965 vtxdumgr0nedg 26978 umgr2wlk 27455 umgrwwlks2on 27463 umgr3cyclex 27712 vdn0conngrumgrv2 27725 |
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