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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 3041 | . . . 4 β’ (πΊ β V β Β¬ πΊ β V) | |
| 2 | ianor 978 | . . . 4 β’ (Β¬ (π β β0 β§ π β 0) β (Β¬ π β β0 β¨ Β¬ π β 0)) | |
| 3 | 1, 2 | orbi12i 911 | . . 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 3041 | . . . . 5 β’ (π β β β Β¬ π β β) | |
| 5 | elnnne0 12487 | . . . . 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 909 | . . 3 β’ ((πΊ β V β¨ π β β) β (πΊ β V β¨ Β¬ (π β β0 β§ π β 0))) |
| 8 | orass 918 | . . 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 978 | . . . . 5 β’ (Β¬ (π β β0 β§ πΊ β V) β (Β¬ π β β0 β¨ Β¬ πΊ β V)) | |
| 11 | orcom 867 | . . . . 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 29783 | . . . . 5 β’ ClWWalksN = (π β β0, π β V β¦ {π€ β (ClWWalksβπ) β£ (β―βπ€) = π}) | |
| 14 | 13 | mpondm0 7643 | . . . 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 2938 | . . . 4 β’ (Β¬ π β 0 β π = 0) | |
| 17 | oveq1 7411 | . . . . 5 β’ (π = 0 β (π ClWWalksN πΊ) = (0 ClWWalksN πΊ)) | |
| 18 | clwwlkn0 29786 | . . . . 5 β’ (0 ClWWalksN πΊ) = β | |
| 19 | 17, 18 | eqtrdi 2782 | . . . 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 854 | . 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 395 β¨ wo 844 = wceq 1533 β wcel 2098 β wne 2934 β wnel 3040 {crab 3426 Vcvv 3468 β c0 4317 βcfv 6536 (class class class)co 7404 0cc0 11109 βcn 12213 β0cn0 12473 β―chash 14293 ClWWalkscclwwlk 29739 ClWWalksN cclwwlkn 29782 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14294 df-word 14469 df-clwwlk 29740 df-clwwlkn 29783 |
| This theorem is referenced by: clwwlknnn 29791 |
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