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Mirrors > Home > MPE Home > Th. List > usgrupgr | Structured version Visualization version GIF version |
Description: A simple graph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 20-Aug-2017.) (Revised by AV, 15-Oct-2020.) |
Ref | Expression |
---|---|
usgrupgr | ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgruspgr 26963 | . 2 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) | |
2 | uspgrupgr 26961 | . 2 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 UPGraphcupgr 26865 USPGraphcuspgr 26933 USGraphcusgr 26934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-i2m1 10605 ax-1ne0 10606 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-2 11701 df-upgr 26867 df-uspgr 26935 df-usgr 26936 |
This theorem is referenced by: usgruhgr 26968 usgredg2vtx 27001 fusgrfupgrfs 27113 cusgr3vnbpr 27218 cusgrres 27230 usgr2wlkneq 27537 usgr2trlncl 27541 usgr2pth 27545 wpthswwlks2on 27740 usgr2wspthon 27744 n4cyclfrgr 28070 |
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