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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 27863, 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 11980 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | 1z 12000 | . . . . . 6 ⊢ 1 ∈ ℤ | |
3 | 1, 2, 1 | 3pm3.2i 1331 | . . . . 5 ⊢ (0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) |
4 | 0ne1 11696 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | wlk2v2e.f | . . . . . . 7 ⊢ 𝐹 = 〈“00”〉 | |
6 | s2prop 14257 | . . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ) → 〈“00”〉 = {〈0, 0〉, 〈1, 0〉}) | |
7 | 1, 1, 6 | mp2an 688 | . . . . . . 7 ⊢ 〈“00”〉 = {〈0, 0〉, 〈1, 0〉} |
8 | 5, 7 | eqtri 2841 | . . . . . 6 ⊢ 𝐹 = {〈0, 0〉, 〈1, 0〉} |
9 | 8 | fpropnf1 7016 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) ∧ 0 ≠ 1) → (Fun 𝐹 ∧ ¬ Fun ◡𝐹)) |
10 | 3, 4, 9 | mp2an 688 | . . . 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 27405 | . 2 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) | |
14 | 12, 13 | mtbir 324 | 1 ⊢ ¬ 𝐹(Trails‘𝐺)𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 {cpr 4559 〈cop 4563 class class class wbr 5057 ◡ccnv 5547 Fun wfun 6342 ‘cfv 6348 0cc0 10525 1c1 10526 ℤcz 11969 〈“cs1 13937 〈“cs2 14191 〈“cs3 14192 Walkscwlks 27305 Trailsctrls 27399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-wlks 27308 df-trls 27401 |
This theorem is referenced by: (None) |
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