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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 3047 | . . . 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 3047 | . . . . 5 β’ (π β β β Β¬ π β β) | |
5 | elnnne0 12435 | . . . . 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 29018 | . . . . 5 β’ ClWWalksN = (π β β0, π β V β¦ {π€ β (ClWWalksβπ) β£ (β―βπ€) = π}) | |
14 | 13 | mpondm0 7598 | . . . 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 2944 | . . . 4 β’ (Β¬ π β 0 β π = 0) | |
17 | oveq1 7368 | . . . . 5 β’ (π = 0 β (π ClWWalksN πΊ) = (0 ClWWalksN πΊ)) | |
18 | clwwlkn0 29021 | . . . . 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 2940 β wnel 3046 {crab 3406 Vcvv 3447 β c0 4286 βcfv 6500 (class class class)co 7361 0cc0 11059 βcn 12161 β0cn0 12421 β―chash 14239 ClWWalkscclwwlk 28974 ClWWalksN cclwwlkn 29017 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-xnn0 12494 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-clwwlk 28975 df-clwwlkn 29018 |
This theorem is referenced by: clwwlknnn 29026 |
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