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Mirrors > Home > MPE Home > Th. List > Mathboxes > prclisacycgr | Structured version Visualization version GIF version |
Description: A proper class (representing a null graph, see vtxvalprc 28284) has the property of an acyclic graph (see also acycgr0v 34076). (Contributed by BTernaryTau, 11-Oct-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
prclisacycgr.1 | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
prclisacycgr | ⊢ (¬ 𝐺 ∈ V → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prclisacycgr.1 | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | fvprc 6879 | . . 3 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅) | |
3 | 1, 2 | eqtrid 2785 | . 2 ⊢ (¬ 𝐺 ∈ V → 𝑉 = ∅) |
4 | br0 5195 | . . . . . 6 ⊢ ¬ 𝑓∅𝑝 | |
5 | df-cycls 29023 | . . . . . . . . . 10 ⊢ Cycles = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}) | |
6 | 5 | relmptopab 7650 | . . . . . . . . 9 ⊢ Rel (Cycles‘𝐺) |
7 | cycliswlk 29034 | . . . . . . . . . 10 ⊢ (𝑓(Cycles‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) | |
8 | df-br 5147 | . . . . . . . . . 10 ⊢ (𝑓(Cycles‘𝐺)𝑝 ↔ 〈𝑓, 𝑝〉 ∈ (Cycles‘𝐺)) | |
9 | df-br 5147 | . . . . . . . . . 10 ⊢ (𝑓(Walks‘𝐺)𝑝 ↔ 〈𝑓, 𝑝〉 ∈ (Walks‘𝐺)) | |
10 | 7, 8, 9 | 3imtr3i 291 | . . . . . . . . 9 ⊢ (〈𝑓, 𝑝〉 ∈ (Cycles‘𝐺) → 〈𝑓, 𝑝〉 ∈ (Walks‘𝐺)) |
11 | 6, 10 | relssi 5784 | . . . . . . . 8 ⊢ (Cycles‘𝐺) ⊆ (Walks‘𝐺) |
12 | 1 | eqeq1i 2738 | . . . . . . . . 9 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
13 | g0wlk0 28888 | . . . . . . . . 9 ⊢ ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅) | |
14 | 12, 13 | sylbi 216 | . . . . . . . 8 ⊢ (𝑉 = ∅ → (Walks‘𝐺) = ∅) |
15 | 11, 14 | sseqtrid 4032 | . . . . . . 7 ⊢ (𝑉 = ∅ → (Cycles‘𝐺) ⊆ ∅) |
16 | ss0 4396 | . . . . . . 7 ⊢ ((Cycles‘𝐺) ⊆ ∅ → (Cycles‘𝐺) = ∅) | |
17 | breq 5148 | . . . . . . . 8 ⊢ ((Cycles‘𝐺) = ∅ → (𝑓(Cycles‘𝐺)𝑝 ↔ 𝑓∅𝑝)) | |
18 | 17 | notbid 318 | . . . . . . 7 ⊢ ((Cycles‘𝐺) = ∅ → (¬ 𝑓(Cycles‘𝐺)𝑝 ↔ ¬ 𝑓∅𝑝)) |
19 | 15, 16, 18 | 3syl 18 | . . . . . 6 ⊢ (𝑉 = ∅ → (¬ 𝑓(Cycles‘𝐺)𝑝 ↔ ¬ 𝑓∅𝑝)) |
20 | 4, 19 | mpbiri 258 | . . . . 5 ⊢ (𝑉 = ∅ → ¬ 𝑓(Cycles‘𝐺)𝑝) |
21 | 20 | intnanrd 491 | . . . 4 ⊢ (𝑉 = ∅ → ¬ (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
22 | 21 | nexdv 1940 | . . 3 ⊢ (𝑉 = ∅ → ¬ ∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
23 | 22 | nexdv 1940 | . 2 ⊢ (𝑉 = ∅ → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
24 | 3, 23 | syl 17 | 1 ⊢ (¬ 𝐺 ∈ V → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ⊆ wss 3946 ∅c0 4320 〈cop 4632 class class class wbr 5146 ‘cfv 6539 0cc0 11105 ♯chash 14285 Vtxcvtx 28235 Walkscwlks 28832 Pathscpths 28948 Cyclesccycls 29021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-fzo 13623 df-hash 14286 df-word 14460 df-wlks 28835 df-trls 28928 df-pths 28952 df-cycls 29023 |
This theorem is referenced by: (None) |
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