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Mirrors > Home > MPE Home > Th. List > usgruhgr | Structured version Visualization version GIF version |
Description: A simple graph is an undirected hypergraph. (Contributed by AV, 9-Feb-2018.) (Revised by AV, 15-Oct-2020.) |
Ref | Expression |
---|---|
usgruhgr | ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrupgr 26961 | . 2 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph) | |
2 | upgruhgr 26881 | . 2 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 UHGraphcuhgr 26835 UPGraphcupgr 26859 USGraphcusgr 26928 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-i2m1 10599 ax-1ne0 10600 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-2 11694 df-uhgr 26837 df-upgr 26861 df-uspgr 26929 df-usgr 26930 |
This theorem is referenced by: usgredg2vtxeuALT 26998 usgr0vb 27013 usgr1vr 27031 subusgr 27065 usgrspan 27071 usgr1v0e 27102 fusgrfisbase 27104 cusgrsize 27230 vtxdusgr0edgnel 27271 usgrvd00 27311 usgr0edg0rusgr 27351 rgrusgrprc 27365 frgr0v 28035 2pthfrgr 28057 |
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