Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 27566 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) | |
2 | relpths 27495 | . . . . . . . . 9 ⊢ Rel (Paths‘𝐺) | |
3 | 2 | brrelex1i 5603 | . . . . . . . 8 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹 ∈ V) |
4 | hasheq0 13718 | . . . . . . . . . 10 ⊢ (𝐹 ∈ V → ((♯‘𝐹) = 0 ↔ 𝐹 = ∅)) | |
5 | 4 | necon3bid 3060 | . . . . . . . . 9 ⊢ (𝐹 ∈ V → ((♯‘𝐹) ≠ 0 ↔ 𝐹 ≠ ∅)) |
6 | 5 | bicomd 225 | . . . . . . . 8 ⊢ (𝐹 ∈ V → (𝐹 ≠ ∅ ↔ (♯‘𝐹) ≠ 0)) |
7 | 3, 6 | syl 17 | . . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝐹 ≠ ∅ ↔ (♯‘𝐹) ≠ 0)) |
8 | 7 | biimpa 479 | . . . . . 6 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≠ 0) |
9 | spthdep 27509 | . . . . . . . 8 ⊢ ((𝐹(SPaths‘𝐺)𝑃 ∧ (♯‘𝐹) ≠ 0) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) | |
10 | 9 | neneqd 3021 | . . . . . . 7 ⊢ ((𝐹(SPaths‘𝐺)𝑃 ∧ (♯‘𝐹) ≠ 0) → ¬ (𝑃‘0) = (𝑃‘(♯‘𝐹))) |
11 | 10 | expcom 416 | . . . . . 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 136 | . . . 4 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ¬ 𝐹(SPaths‘𝐺)𝑃)) |
14 | 13 | impancom 454 | . . 3 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝐹 ≠ ∅ → ¬ 𝐹(SPaths‘𝐺)𝑃)) |
15 | 1, 14 | sylbi 219 | . 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 ∅c0 4291 class class class wbr 5059 ‘cfv 6350 0cc0 10531 ♯chash 13684 Pathscpths 27487 SPathscspths 27488 Cyclesccycls 27560 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-wlks 27375 df-trls 27468 df-pths 27491 df-spths 27492 df-cycls 27562 |
This theorem is referenced by: spthcycl 32371 |
Copyright terms: Public domain | W3C validator |