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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 28573) has the property of an acyclic graph (see also acycgr0v 34438). (Contributed by BTernaryTau, 11-Oct-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
prclisacycgr.1 | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
prclisacycgr | β’ (Β¬ πΊ β V β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prclisacycgr.1 | . . 3 β’ π = (VtxβπΊ) | |
2 | fvprc 6883 | . . 3 β’ (Β¬ πΊ β V β (VtxβπΊ) = β ) | |
3 | 1, 2 | eqtrid 2783 | . 2 β’ (Β¬ πΊ β V β π = β ) |
4 | br0 5197 | . . . . . 6 β’ Β¬ πβ π | |
5 | df-cycls 29312 | . . . . . . . . . 10 β’ Cycles = (π β V β¦ {β¨π, πβ© β£ (π(Pathsβπ)π β§ (πβ0) = (πβ(β―βπ)))}) | |
6 | 5 | relmptopab 7660 | . . . . . . . . 9 β’ Rel (CyclesβπΊ) |
7 | cycliswlk 29323 | . . . . . . . . . 10 β’ (π(CyclesβπΊ)π β π(WalksβπΊ)π) | |
8 | df-br 5149 | . . . . . . . . . 10 β’ (π(CyclesβπΊ)π β β¨π, πβ© β (CyclesβπΊ)) | |
9 | df-br 5149 | . . . . . . . . . 10 β’ (π(WalksβπΊ)π β β¨π, πβ© β (WalksβπΊ)) | |
10 | 7, 8, 9 | 3imtr3i 291 | . . . . . . . . 9 β’ (β¨π, πβ© β (CyclesβπΊ) β β¨π, πβ© β (WalksβπΊ)) |
11 | 6, 10 | relssi 5787 | . . . . . . . 8 β’ (CyclesβπΊ) β (WalksβπΊ) |
12 | 1 | eqeq1i 2736 | . . . . . . . . 9 β’ (π = β β (VtxβπΊ) = β ) |
13 | g0wlk0 29177 | . . . . . . . . 9 β’ ((VtxβπΊ) = β β (WalksβπΊ) = β ) | |
14 | 12, 13 | sylbi 216 | . . . . . . . 8 β’ (π = β β (WalksβπΊ) = β ) |
15 | 11, 14 | sseqtrid 4034 | . . . . . . 7 β’ (π = β β (CyclesβπΊ) β β ) |
16 | ss0 4398 | . . . . . . 7 β’ ((CyclesβπΊ) β β β (CyclesβπΊ) = β ) | |
17 | breq 5150 | . . . . . . . 8 β’ ((CyclesβπΊ) = β β (π(CyclesβπΊ)π β πβ π)) | |
18 | 17 | notbid 318 | . . . . . . 7 β’ ((CyclesβπΊ) = β β (Β¬ π(CyclesβπΊ)π β Β¬ πβ π)) |
19 | 15, 16, 18 | 3syl 18 | . . . . . 6 β’ (π = β β (Β¬ π(CyclesβπΊ)π β Β¬ πβ π)) |
20 | 4, 19 | mpbiri 258 | . . . . 5 β’ (π = β β Β¬ π(CyclesβπΊ)π) |
21 | 20 | intnanrd 489 | . . . 4 β’ (π = β β Β¬ (π(CyclesβπΊ)π β§ π β β )) |
22 | 21 | nexdv 1938 | . . 3 β’ (π = β β Β¬ βπ(π(CyclesβπΊ)π β§ π β β )) |
23 | 22 | nexdv 1938 | . 2 β’ (π = β β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β )) |
24 | 3, 23 | syl 17 | 1 β’ (Β¬ πΊ β V β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 = wceq 1540 βwex 1780 β wcel 2105 β wne 2939 Vcvv 3473 β wss 3948 β c0 4322 β¨cop 4634 class class class wbr 5148 βcfv 6543 0cc0 11114 β―chash 14295 Vtxcvtx 28524 Walkscwlks 29121 Pathscpths 29237 Cyclesccycls 29310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-wlks 29124 df-trls 29217 df-pths 29241 df-cycls 29312 |
This theorem is referenced by: (None) |
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