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Mirrors > Home > MPE Home > Th. List > ntrl2v2e | Structured version Visualization version GIF version |
Description: A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk, see wlk2v2e 28507, but not a trail. Notice that 𝐺 is a simple graph (without loops) only if 𝑋 ≠ 𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
wlk2v2e.i | ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 |
wlk2v2e.f | ⊢ 𝐹 = 〈“00”〉 |
wlk2v2e.x | ⊢ 𝑋 ∈ V |
wlk2v2e.y | ⊢ 𝑌 ∈ V |
wlk2v2e.p | ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 |
wlk2v2e.g | ⊢ 𝐺 = 〈{𝑋, 𝑌}, 𝐼〉 |
Ref | Expression |
---|---|
ntrl2v2e | ⊢ ¬ 𝐹(Trails‘𝐺)𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 12318 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | 1z 12338 | . . . . . 6 ⊢ 1 ∈ ℤ | |
3 | 1, 2, 1 | 3pm3.2i 1338 | . . . . 5 ⊢ (0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) |
4 | 0ne1 12032 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | wlk2v2e.f | . . . . . . 7 ⊢ 𝐹 = 〈“00”〉 | |
6 | s2prop 14608 | . . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ) → 〈“00”〉 = {〈0, 0〉, 〈1, 0〉}) | |
7 | 1, 1, 6 | mp2an 689 | . . . . . . 7 ⊢ 〈“00”〉 = {〈0, 0〉, 〈1, 0〉} |
8 | 5, 7 | eqtri 2766 | . . . . . 6 ⊢ 𝐹 = {〈0, 0〉, 〈1, 0〉} |
9 | 8 | fpropnf1 7133 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) ∧ 0 ≠ 1) → (Fun 𝐹 ∧ ¬ Fun ◡𝐹)) |
10 | 3, 4, 9 | mp2an 689 | . . . 4 ⊢ (Fun 𝐹 ∧ ¬ Fun ◡𝐹) |
11 | 10 | simpri 486 | . . 3 ⊢ ¬ Fun ◡𝐹 |
12 | 11 | intnan 487 | . 2 ⊢ ¬ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹) |
13 | istrl 28051 | . 2 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) | |
14 | 12, 13 | mtbir 323 | 1 ⊢ ¬ 𝐹(Trails‘𝐺)𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3430 {cpr 4564 〈cop 4568 class class class wbr 5074 ◡ccnv 5584 Fun wfun 6421 ‘cfv 6427 0cc0 10859 1c1 10860 ℤcz 12307 〈“cs1 14288 〈“cs2 14542 〈“cs3 14543 Walkscwlks 27951 Trailsctrls 28045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
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 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-n0 12222 df-z 12308 df-uz 12571 df-fz 13228 df-fzo 13371 df-hash 14033 df-word 14206 df-concat 14262 df-s1 14289 df-s2 14549 df-wlks 27954 df-trls 28047 |
This theorem is referenced by: (None) |
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