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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 27017, 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 11388 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | 1z 11407 | . . . . . 6 ⊢ 1 ∈ ℤ | |
3 | 1, 2, 1 | 3pm3.2i 1239 | . . . . 5 ⊢ (0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) |
4 | 0ne1 11088 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | wlk2v2e.f | . . . . . . 7 ⊢ 𝐹 = 〈“00”〉 | |
6 | s2prop 13652 | . . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ) → 〈“00”〉 = {〈0, 0〉, 〈1, 0〉}) | |
7 | 1, 1, 6 | mp2an 708 | . . . . . . 7 ⊢ 〈“00”〉 = {〈0, 0〉, 〈1, 0〉} |
8 | 5, 7 | eqtri 2644 | . . . . . 6 ⊢ 𝐹 = {〈0, 0〉, 〈1, 0〉} |
9 | 8 | fpropnf1 6524 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) ∧ 0 ≠ 1) → (Fun 𝐹 ∧ ¬ Fun ◡𝐹)) |
10 | 3, 4, 9 | mp2an 708 | . . . 4 ⊢ (Fun 𝐹 ∧ ¬ Fun ◡𝐹) |
11 | 10 | simpri 478 | . . 3 ⊢ ¬ Fun ◡𝐹 |
12 | 11 | intnan 960 | . 2 ⊢ ¬ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹) |
13 | istrl 26593 | . 2 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) | |
14 | 12, 13 | mtbir 313 | 1 ⊢ ¬ 𝐹(Trails‘𝐺)𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 {cpr 4179 〈cop 4183 class class class wbr 4653 ◡ccnv 5113 Fun wfun 5882 ‘cfv 5888 0cc0 9936 1c1 9937 ℤcz 11377 〈“cs1 13294 〈“cs2 13586 〈“cs3 13587 Walkscwlks 26492 Trailsctrls 26587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-wlks 26495 df-trls 26589 |
This theorem is referenced by: (None) |
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