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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 1wlk2v2e 41324, 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.) |
Ref | Expression |
---|---|
1wlk2v2e.i | ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 |
1wlk2v2e.f | ⊢ 𝐹 = 〈“00”〉 |
1wlk2v2e.x | ⊢ 𝑋 ∈ V |
1wlk2v2e.y | ⊢ 𝑌 ∈ V |
1wlk2v2e.p | ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 |
1wlk2v2e.g | ⊢ 𝐺 = 〈{𝑋, 𝑌}, 𝐼〉 |
Ref | Expression |
---|---|
ntrl2v2e | ⊢ ¬ 𝐹(TrailS‘𝐺)𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11265 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | 1z 11284 | . . . . . 6 ⊢ 1 ∈ ℤ | |
3 | 1, 2, 1 | 3pm3.2i 1232 | . . . . 5 ⊢ (0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) |
4 | 0ne1 10965 | . . . . 5 ⊢ 0 ≠ 1 | |
5 | 1wlk2v2e.f | . . . . . . 7 ⊢ 𝐹 = 〈“00”〉 | |
6 | s2prop 13502 | . . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ 0 ∈ ℤ) → 〈“00”〉 = {〈0, 0〉, 〈1, 0〉}) | |
7 | 1, 1, 6 | mp2an 704 | . . . . . . 7 ⊢ 〈“00”〉 = {〈0, 0〉, 〈1, 0〉} |
8 | 5, 7 | eqtri 2632 | . . . . . 6 ⊢ 𝐹 = {〈0, 0〉, 〈1, 0〉} |
9 | 8 | fpropnf1 40337 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ∈ ℤ) ∧ 0 ≠ 1) → (Fun 𝐹 ∧ ¬ Fun ◡𝐹)) |
10 | 3, 4, 9 | mp2an 704 | . . . 4 ⊢ (Fun 𝐹 ∧ ¬ Fun ◡𝐹) |
11 | 10 | simpri 477 | . . 3 ⊢ ¬ Fun ◡𝐹 |
12 | 11 | intnan 951 | . 2 ⊢ ¬ (𝐹(1Walks‘𝐺)𝑃 ∧ Fun ◡𝐹) |
13 | 1wlk2v2e.g | . . . 4 ⊢ 𝐺 = 〈{𝑋, 𝑌}, 𝐼〉 | |
14 | opex 4859 | . . . 4 ⊢ 〈{𝑋, 𝑌}, 𝐼〉 ∈ V | |
15 | 13, 14 | eqeltri 2684 | . . 3 ⊢ 𝐺 ∈ V |
16 | s2cli 13475 | . . . 4 ⊢ 〈“00”〉 ∈ Word V | |
17 | 5, 16 | eqeltri 2684 | . . 3 ⊢ 𝐹 ∈ Word V |
18 | 1wlk2v2e.p | . . . 4 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
19 | s3cli 13476 | . . . 4 ⊢ 〈“𝑋𝑌𝑋”〉 ∈ Word V | |
20 | 18, 19 | eqeltri 2684 | . . 3 ⊢ 𝑃 ∈ Word V |
21 | isTrl 40904 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) → (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))) | |
22 | 15, 17, 20, 21 | mp3an 1416 | . 2 ⊢ (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) |
23 | 12, 22 | mtbir 312 | 1 ⊢ ¬ 𝐹(TrailS‘𝐺)𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 {cpr 4127 〈cop 4131 class class class wbr 4583 ◡ccnv 5037 Fun wfun 5798 ‘cfv 5804 0cc0 9815 1c1 9816 ℤcz 11254 Word cword 13146 〈“cs1 13149 〈“cs2 13437 〈“cs3 13438 1Walksc1wlks 40796 TrailSctrls 40899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-1wlks 40800 df-trls 40901 |
This theorem is referenced by: (None) |
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