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Mirrors > Home > MPE Home > Th. List > cyclnspth | Structured version Visualization version GIF version |
Description: A (non-trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
cyclnspth | ⊢ (𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscycl 29677 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | |
2 | relpths 29606 | . . . . . . . . 9 ⊢ Rel (Paths‘𝐺) | |
3 | 2 | brrelex1i 5734 | . . . . . . . 8 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹 ∈ V) |
4 | hasheq0 14358 | . . . . . . . . . 10 ⊢ (𝐹 ∈ V → ((♯‘𝐹) = 0 ↔ 𝐹 = ∅)) | |
5 | 4 | necon3bid 2974 | . . . . . . . . 9 ⊢ (𝐹 ∈ V → ((♯‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅)) |
6 | 5 | bicomd 222 | . . . . . . . 8 ⊢ (𝐹 ∈ V → (𝐹 ≠ ∅ ↔ (♯‘𝐹) ≠ 0)) |
7 | 3, 6 | syl 17 | . . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝐹 ≠ ∅ ↔ (♯‘𝐹) ≠ 0)) |
8 | 7 | biimpa 475 | . . . . . 6 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≠ 0) |
9 | spthdep 29620 | . . . . . . . 8 ⊢ ((𝐹(SPaths‘𝐺)𝑃 ∧ (♯‘𝐹) ≠ 0) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) | |
10 | 9 | neneqd 2934 | . . . . . . 7 ⊢ ((𝐹(SPaths‘𝐺)𝑃 ∧ (♯‘𝐹) ≠ 0) → ¬ (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
11 | 10 | expcom 412 | . . . . . 6 ⊢ ((♯‘𝐹) ≠ 0 → (𝐹(SPaths‘𝐺)𝑃 → ¬ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) |
12 | 8, 11 | syl 17 | . . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (𝐹(SPaths‘𝐺)𝑃 → ¬ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) |
13 | 12 | con2d 134 | . . . 4 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ¬ 𝐹(SPaths‘𝐺)𝑃)) |
14 | 13 | impancom 450 | . . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝐹 ≠ ∅ → ¬ 𝐹(SPaths‘𝐺)𝑃)) |
15 | 1, 14 | sylbi 216 | . 2 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (𝐹 ≠ ∅ → ¬ 𝐹(SPaths‘𝐺)𝑃)) |
16 | 15 | com12 32 | 1 ⊢ (𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ∅c0 4322 class class class wbr 5149 ‘cfv 6549 0cc0 11140 ♯chash 14325 Pathscpths 29598 SPathscspths 29599 Cyclesccycls 29671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-wlks 29485 df-trls 29578 df-pths 29602 df-spths 29603 df-cycls 29673 |
This theorem is referenced by: spthcycl 34867 |
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