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| Mirrors > Home > MPE Home > Th. List > clwwlkneq0 | Structured version Visualization version GIF version | ||
| Description: Sufficient conditions for ClWWalksN to be empty. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 24-Feb-2022.) |
| Ref | Expression |
|---|---|
| clwwlkneq0 | β’ ((πΊ β V β¨ π β β) β (π ClWWalksN πΊ) = β ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-nel 3048 | . . . 4 β’ (πΊ β V β Β¬ πΊ β V) | |
| 2 | ianor 981 | . . . 4 β’ (Β¬ (π β β0 β§ π β 0) β (Β¬ π β β0 β¨ Β¬ π β 0)) | |
| 3 | 1, 2 | orbi12i 914 | . . 3 β’ ((πΊ β V β¨ Β¬ (π β β0 β§ π β 0)) β (Β¬ πΊ β V β¨ (Β¬ π β β0 β¨ Β¬ π β 0))) |
| 4 | df-nel 3048 | . . . . 5 β’ (π β β β Β¬ π β β) | |
| 5 | elnnne0 12486 | . . . . 5 β’ (π β β β (π β β0 β§ π β 0)) | |
| 6 | 4, 5 | xchbinx 334 | . . . 4 β’ (π β β β Β¬ (π β β0 β§ π β 0)) |
| 7 | 6 | orbi2i 912 | . . 3 β’ ((πΊ β V β¨ π β β) β (πΊ β V β¨ Β¬ (π β β0 β§ π β 0))) |
| 8 | orass 921 | . . 3 β’ (((Β¬ πΊ β V β¨ Β¬ π β β0) β¨ Β¬ π β 0) β (Β¬ πΊ β V β¨ (Β¬ π β β0 β¨ Β¬ π β 0))) | |
| 9 | 3, 7, 8 | 3bitr4i 303 | . 2 β’ ((πΊ β V β¨ π β β) β ((Β¬ πΊ β V β¨ Β¬ π β β0) β¨ Β¬ π β 0)) |
| 10 | ianor 981 | . . . . 5 β’ (Β¬ (π β β0 β§ πΊ β V) β (Β¬ π β β0 β¨ Β¬ πΊ β V)) | |
| 11 | orcom 869 | . . . . 5 β’ ((Β¬ π β β0 β¨ Β¬ πΊ β V) β (Β¬ πΊ β V β¨ Β¬ π β β0)) | |
| 12 | 10, 11 | bitri 275 | . . . 4 β’ (Β¬ (π β β0 β§ πΊ β V) β (Β¬ πΊ β V β¨ Β¬ π β β0)) |
| 13 | df-clwwlkn 29278 | . . . . 5 β’ ClWWalksN = (π β β0, π β V β¦ {π€ β (ClWWalksβπ) β£ (β―βπ€) = π}) | |
| 14 | 13 | mpondm0 7647 | . . . 4 β’ (Β¬ (π β β0 β§ πΊ β V) β (π ClWWalksN πΊ) = β ) |
| 15 | 12, 14 | sylbir 234 | . . 3 β’ ((Β¬ πΊ β V β¨ Β¬ π β β0) β (π ClWWalksN πΊ) = β ) |
| 16 | nne 2945 | . . . 4 β’ (Β¬ π β 0 β π = 0) | |
| 17 | oveq1 7416 | . . . . 5 β’ (π = 0 β (π ClWWalksN πΊ) = (0 ClWWalksN πΊ)) | |
| 18 | clwwlkn0 29281 | . . . . 5 β’ (0 ClWWalksN πΊ) = β | |
| 19 | 17, 18 | eqtrdi 2789 | . . . 4 β’ (π = 0 β (π ClWWalksN πΊ) = β ) |
| 20 | 16, 19 | sylbi 216 | . . 3 β’ (Β¬ π β 0 β (π ClWWalksN πΊ) = β ) |
| 21 | 15, 20 | jaoi 856 | . 2 β’ (((Β¬ πΊ β V β¨ Β¬ π β β0) β¨ Β¬ π β 0) β (π ClWWalksN πΊ) = β ) |
| 22 | 9, 21 | sylbi 216 | 1 β’ ((πΊ β V β¨ π β β) β (π ClWWalksN πΊ) = β ) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β¨ wo 846 = wceq 1542 β wcel 2107 β wne 2941 β wnel 3047 {crab 3433 Vcvv 3475 β c0 4323 βcfv 6544 (class class class)co 7409 0cc0 11110 βcn 12212 β0cn0 12472 β―chash 14290 ClWWalkscclwwlk 29234 ClWWalksN cclwwlkn 29277 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-clwwlk 29235 df-clwwlkn 29278 |
| This theorem is referenced by: clwwlknnn 29286 |
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