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Mirrors > Home > MPE Home > Th. List > usgr2trlncrct | Structured version Visualization version GIF version |
Description: In a simple graph, any trail of length 2 is not a circuit. (Contributed by AV, 5-Jun-2021.) |
Ref | Expression |
---|---|
usgr2trlncrct | ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → ¬ 𝐹(Circuits‘𝐺)𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr2trlncl 28137 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2))) | |
2 | 1 | imp 407 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ 𝐹(Trails‘𝐺)𝑃) → (𝑃‘0) ≠ (𝑃‘2)) |
3 | crctprop 28169 | . . . . . . 7 ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | |
4 | fveq2 6771 | . . . . . . . . 9 ⊢ ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = (𝑃‘2)) | |
5 | 4 | eqeq2d 2751 | . . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) = (𝑃‘2))) |
6 | 5 | biimpcd 248 | . . . . . . 7 ⊢ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ((♯‘𝐹) = 2 → (𝑃‘0) = (𝑃‘2))) |
7 | 3, 6 | simpl2im 504 | . . . . . 6 ⊢ (𝐹(Circuits‘𝐺)𝑃 → ((♯‘𝐹) = 2 → (𝑃‘0) = (𝑃‘2))) |
8 | 7 | com12 32 | . . . . 5 ⊢ ((♯‘𝐹) = 2 → (𝐹(Circuits‘𝐺)𝑃 → (𝑃‘0) = (𝑃‘2))) |
9 | 8 | ad2antlr 724 | . . . 4 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ 𝐹(Trails‘𝐺)𝑃) → (𝐹(Circuits‘𝐺)𝑃 → (𝑃‘0) = (𝑃‘2))) |
10 | 9 | necon3ad 2958 | . . 3 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ 𝐹(Trails‘𝐺)𝑃) → ((𝑃‘0) ≠ (𝑃‘2) → ¬ 𝐹(Circuits‘𝐺)𝑃)) |
11 | 2, 10 | mpd 15 | . 2 ⊢ (((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) ∧ 𝐹(Trails‘𝐺)𝑃) → ¬ 𝐹(Circuits‘𝐺)𝑃) |
12 | 11 | ex 413 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → ¬ 𝐹(Circuits‘𝐺)𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 class class class wbr 5079 ‘cfv 6432 0cc0 10882 2c2 12039 ♯chash 14055 USGraphcusgr 27530 Trailsctrls 28068 Circuitsccrcts 28161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-oadd 8293 df-er 8490 df-map 8609 df-pm 8610 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-dju 9670 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-n0 12245 df-xnn0 12317 df-z 12331 df-uz 12594 df-fz 13251 df-fzo 13394 df-hash 14056 df-word 14229 df-edg 27429 df-uhgr 27439 df-upgr 27463 df-uspgr 27531 df-usgr 27532 df-wlks 27977 df-trls 28070 df-crcts 28163 |
This theorem is referenced by: (None) |
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