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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 3044 | . . . 4 β’ (πΊ β V β Β¬ πΊ β V) | |
| 2 | ianor 980 | . . . 4 β’ (Β¬ (π β β0 β§ π β 0) β (Β¬ π β β0 β¨ Β¬ π β 0)) | |
| 3 | 1, 2 | orbi12i 913 | . . 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 3044 | . . . . 5 β’ (π β β β Β¬ π β β) | |
| 5 | elnnne0 12517 | . . . . 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 911 | . . 3 β’ ((πΊ β V β¨ π β β) β (πΊ β V β¨ Β¬ (π β β0 β§ π β 0))) |
| 8 | orass 920 | . . 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 980 | . . . . 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 29848 | . . . . 5 β’ ClWWalksN = (π β β0, π β V β¦ {π€ β (ClWWalksβπ) β£ (β―βπ€) = π}) | |
| 14 | 13 | mpondm0 7661 | . . . 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 2941 | . . . 4 β’ (Β¬ π β 0 β π = 0) | |
| 17 | oveq1 7427 | . . . . 5 β’ (π = 0 β (π ClWWalksN πΊ) = (0 ClWWalksN πΊ)) | |
| 18 | clwwlkn0 29851 | . . . . 5 β’ (0 ClWWalksN πΊ) = β | |
| 19 | 17, 18 | eqtrdi 2784 | . . . 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 395 β¨ wo 846 = wceq 1534 β wcel 2099 β wne 2937 β wnel 3043 {crab 3429 Vcvv 3471 β c0 4323 βcfv 6548 (class class class)co 7420 0cc0 11139 βcn 12243 β0cn0 12503 β―chash 14322 ClWWalkscclwwlk 29804 ClWWalksN cclwwlkn 29847 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-xnn0 12576 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-clwwlk 29805 df-clwwlkn 29848 |
| This theorem is referenced by: clwwlknnn 29856 |
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