![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > clwwlknon1nloop | Structured version Visualization version GIF version |
Description: If there is no loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is empty. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknon1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
clwwlknon1.c | ⊢ 𝐶 = (ClWWalksNOn‘𝐺) |
clwwlknon1.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
clwwlknon1nloop | ⊢ ({𝑋} ∉ 𝐸 → (𝑋𝐶1) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlknon1.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | clwwlknon1.c | . . . . 5 ⊢ 𝐶 = (ClWWalksNOn‘𝐺) | |
3 | clwwlknon1.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
4 | 1, 2, 3 | clwwlknon1 29900 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)}) |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)}) |
6 | df-nel 3042 | . . . . . . . . 9 ⊢ ({𝑋} ∉ 𝐸 ↔ ¬ {𝑋} ∈ 𝐸) | |
7 | 6 | biimpi 215 | . . . . . . . 8 ⊢ ({𝑋} ∉ 𝐸 → ¬ {𝑋} ∈ 𝐸) |
8 | 7 | olcd 873 | . . . . . . 7 ⊢ ({𝑋} ∉ 𝐸 → (¬ 𝑤 = 〈“𝑋”〉 ∨ ¬ {𝑋} ∈ 𝐸)) |
9 | 8 | ad2antlr 726 | . . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) ∧ 𝑤 ∈ Word 𝑉) → (¬ 𝑤 = 〈“𝑋”〉 ∨ ¬ {𝑋} ∈ 𝐸)) |
10 | ianor 980 | . . . . . 6 ⊢ (¬ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸) ↔ (¬ 𝑤 = 〈“𝑋”〉 ∨ ¬ {𝑋} ∈ 𝐸)) | |
11 | 9, 10 | sylibr 233 | . . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) ∧ 𝑤 ∈ Word 𝑉) → ¬ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) |
12 | 11 | ralrimiva 3141 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → ∀𝑤 ∈ Word 𝑉 ¬ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) |
13 | rabeq0 4380 | . . . 4 ⊢ ({𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)} = ∅ ↔ ∀𝑤 ∈ Word 𝑉 ¬ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)) | |
14 | 12, 13 | sylibr 233 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)} = ∅) |
15 | 5, 14 | eqtrd 2767 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → (𝑋𝐶1) = ∅) |
16 | 2 | oveqi 7427 | . . . 4 ⊢ (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1) |
17 | 1 | eleq2i 2820 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺)) |
18 | 17 | notbii 320 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ 𝑉 ↔ ¬ 𝑋 ∈ (Vtx‘𝐺)) |
19 | 18 | biimpi 215 | . . . . . 6 ⊢ (¬ 𝑋 ∈ 𝑉 → ¬ 𝑋 ∈ (Vtx‘𝐺)) |
20 | 19 | intnanrd 489 | . . . . 5 ⊢ (¬ 𝑋 ∈ 𝑉 → ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ)) |
21 | clwwlknon0 29896 | . . . . 5 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
23 | 16, 22 | eqtrid 2779 | . . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝑋𝐶1) = ∅) |
24 | 23 | adantr 480 | . 2 ⊢ ((¬ 𝑋 ∈ 𝑉 ∧ {𝑋} ∉ 𝐸) → (𝑋𝐶1) = ∅) |
25 | 15, 24 | pm2.61ian 811 | 1 ⊢ ({𝑋} ∉ 𝐸 → (𝑋𝐶1) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∉ wnel 3041 ∀wral 3056 {crab 3427 ∅c0 4318 {csn 4624 ‘cfv 6542 (class class class)co 7414 1c1 11133 ℕcn 12236 Word cword 14490 〈“cs1 14571 Vtxcvtx 28802 Edgcedg 28853 ClWWalksNOncclwwlknon 29890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-xnn0 12569 df-z 12583 df-uz 12847 df-fz 13511 df-fzo 13654 df-hash 14316 df-word 14491 df-lsw 14539 df-s1 14572 df-clwwlk 29785 df-clwwlkn 29828 df-clwwlknon 29891 |
This theorem is referenced by: clwwlknon1sn 29903 clwwlknon1le1 29904 |
Copyright terms: Public domain | W3C validator |