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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 30186, 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 12622 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | 1z 12645 | . . . . . 6 ⊢ 1 ∈ ℤ | |
3 | 1, 2, 1 | 3pm3.2i 1338 | . . . . 5 ⊢ (0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) |
4 | 0ne1 12335 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | wlk2v2e.f | . . . . . . 7 ⊢ 𝐹 = 〈“00”〉 | |
6 | s2prop 14943 | . . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ) → 〈“00”〉 = {〈0, 0〉, 〈1, 0〉}) | |
7 | 1, 1, 6 | mp2an 692 | . . . . . . 7 ⊢ 〈“00”〉 = {〈0, 0〉, 〈1, 0〉} |
8 | 5, 7 | eqtri 2763 | . . . . . 6 ⊢ 𝐹 = {〈0, 0〉, 〈1, 0〉} |
9 | 8 | fpropnf1 7287 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) ∧ 0 ≠ 1) → (Fun 𝐹 ∧ ¬ Fun ◡𝐹)) |
10 | 3, 4, 9 | mp2an 692 | . . . 4 ⊢ (Fun 𝐹 ∧ ¬ Fun ◡𝐹) |
11 | 10 | simpri 485 | . . 3 ⊢ ¬ Fun ◡𝐹 |
12 | 11 | intnan 486 | . 2 ⊢ ¬ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹) |
13 | istrl 29729 | . 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 {cpr 4633 〈cop 4637 class class class wbr 5148 ◡ccnv 5688 Fun wfun 6557 ‘cfv 6563 0cc0 11153 1c1 11154 ℤcz 12611 〈“cs1 14630 〈“cs2 14877 〈“cs3 14878 Walkscwlks 29629 Trailsctrls 29723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-wlks 29632 df-trls 29725 |
This theorem is referenced by: (None) |
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