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| Mirrors > Home > MPE Home > Th. List > 0pth | Structured version Visualization version GIF version | ||
| Description: A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 19-Jan-2021.) (Revised by AV, 30-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| 0pth.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| Ref | Expression | 
|---|---|
| 0pth | ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ispth 29741 | . . 3 ⊢ (∅(Paths‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅)) | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅))) | 
| 3 | 3anass 1095 | . . . 4 ⊢ ((∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) ↔ (∅(Trails‘𝐺)𝑃 ∧ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅))) | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝐺 ∈ 𝑊 → ((∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) ↔ (∅(Trails‘𝐺)𝑃 ∧ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅)))) | 
| 5 | funcnv0 6632 | . . . . . 6 ⊢ Fun ◡∅ | |
| 6 | hash0 14406 | . . . . . . . . . . . 12 ⊢ (♯‘∅) = 0 | |
| 7 | 0le1 11786 | . . . . . . . . . . . 12 ⊢ 0 ≤ 1 | |
| 8 | 6, 7 | eqbrtri 5164 | . . . . . . . . . . 11 ⊢ (♯‘∅) ≤ 1 | 
| 9 | 1z 12647 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℤ | |
| 10 | 0z 12624 | . . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ | |
| 11 | 6, 10 | eqeltri 2837 | . . . . . . . . . . . 12 ⊢ (♯‘∅) ∈ ℤ | 
| 12 | fzon 13720 | . . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘∅) ∈ ℤ) → ((♯‘∅) ≤ 1 ↔ (1..^(♯‘∅)) = ∅)) | |
| 13 | 9, 11, 12 | mp2an 692 | . . . . . . . . . . 11 ⊢ ((♯‘∅) ≤ 1 ↔ (1..^(♯‘∅)) = ∅) | 
| 14 | 8, 13 | mpbi 230 | . . . . . . . . . 10 ⊢ (1..^(♯‘∅)) = ∅ | 
| 15 | 14 | reseq2i 5994 | . . . . . . . . 9 ⊢ (𝑃 ↾ (1..^(♯‘∅))) = (𝑃 ↾ ∅) | 
| 16 | res0 6001 | . . . . . . . . 9 ⊢ (𝑃 ↾ ∅) = ∅ | |
| 17 | 15, 16 | eqtri 2765 | . . . . . . . 8 ⊢ (𝑃 ↾ (1..^(♯‘∅))) = ∅ | 
| 18 | 17 | cnveqi 5885 | . . . . . . 7 ⊢ ◡(𝑃 ↾ (1..^(♯‘∅))) = ◡∅ | 
| 19 | 18 | funeqi 6587 | . . . . . 6 ⊢ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ↔ Fun ◡∅) | 
| 20 | 5, 19 | mpbir 231 | . . . . 5 ⊢ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) | 
| 21 | 14 | imaeq2i 6076 | . . . . . . . 8 ⊢ (𝑃 “ (1..^(♯‘∅))) = (𝑃 “ ∅) | 
| 22 | ima0 6095 | . . . . . . . 8 ⊢ (𝑃 “ ∅) = ∅ | |
| 23 | 21, 22 | eqtri 2765 | . . . . . . 7 ⊢ (𝑃 “ (1..^(♯‘∅))) = ∅ | 
| 24 | 23 | ineq2i 4217 | . . . . . 6 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ((𝑃 “ {0, (♯‘∅)}) ∩ ∅) | 
| 25 | in0 4395 | . . . . . 6 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ ∅) = ∅ | |
| 26 | 24, 25 | eqtri 2765 | . . . . 5 ⊢ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅ | 
| 27 | 20, 26 | pm3.2i 470 | . . . 4 ⊢ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) | 
| 28 | 27 | biantru 529 | . . 3 ⊢ (∅(Trails‘𝐺)𝑃 ↔ (∅(Trails‘𝐺)𝑃 ∧ (Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅))) | 
| 29 | 4, 28 | bitr4di 289 | . 2 ⊢ (𝐺 ∈ 𝑊 → ((∅(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘∅))) ∧ ((𝑃 “ {0, (♯‘∅)}) ∩ (𝑃 “ (1..^(♯‘∅)))) = ∅) ↔ ∅(Trails‘𝐺)𝑃)) | 
| 30 | 0pth.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 31 | 30 | 0trl 30141 | . 2 ⊢ (𝐺 ∈ 𝑊 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | 
| 32 | 2, 29, 31 | 3bitrd 305 | 1 ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ∅c0 4333 {cpr 4628 class class class wbr 5143 ◡ccnv 5684 ↾ cres 5687 “ cima 5688 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ≤ cle 11296 ℤcz 12613 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Vtxcvtx 29013 Trailsctrls 29708 Pathscpths 29730 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-wlks 29617 df-trls 29710 df-pths 29734 | 
| This theorem is referenced by: 0pthon 30146 0cycl 30153 | 
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