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Mirrors > Home > MPE Home > Th. List > 2wspiundisj | Structured version Visualization version GIF version |
Description: All simple paths of length 2 from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 5-Mar-2018.) (Revised by AV, 14-May-2021.) (Proof shortened by AV, 9-Jan-2022.) |
Ref | Expression |
---|---|
2wspiundisj | ⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6740 | . . 3 ⊢ (𝑎 = 𝑐 → (𝑎(2 WSPathsNOn 𝐺)𝑏) = (𝑐(2 WSPathsNOn 𝐺)𝑏)) | |
2 | oveq2 6741 | . . 3 ⊢ (𝑏 = 𝑑 → (𝑐(2 WSPathsNOn 𝐺)𝑏) = (𝑐(2 WSPathsNOn 𝐺)𝑑)) | |
3 | sneq 4263 | . . . 4 ⊢ (𝑎 = 𝑐 → {𝑎} = {𝑐}) | |
4 | 3 | difeq2d 3804 | . . 3 ⊢ (𝑎 = 𝑐 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑐})) |
5 | wspthneq1eq2 26858 | . . . . 5 ⊢ ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑)) → (𝑎 = 𝑐 ∧ 𝑏 = 𝑑)) | |
6 | 5 | simpld 477 | . . . 4 ⊢ ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑)) → 𝑎 = 𝑐) |
7 | 6 | 3adant1 1122 | . . 3 ⊢ ((⊤ ∧ 𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑)) → 𝑎 = 𝑐) |
8 | 1, 2, 4, 7 | disjiund 4719 | . 2 ⊢ (⊤ → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) |
9 | 8 | trud 1574 | 1 ⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ⊤wtru 1565 ∈ wcel 2071 ∖ cdif 3645 {csn 4253 ∪ ciun 4596 Disj wdisj 4696 (class class class)co 6733 2c2 11151 WSPathsNOn cwwspthsnon 26821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-rep 4847 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1051 df-3or 1073 df-3an 1074 df-tru 1567 df-fal 1570 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-int 4552 df-iun 4598 df-disj 4697 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-om 7151 df-1st 7253 df-2nd 7254 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-1o 7648 df-oadd 7652 df-er 7830 df-map 7944 df-pm 7945 df-en 8041 df-dom 8042 df-sdom 8043 df-fin 8044 df-card 8846 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-nn 11102 df-n0 11374 df-z 11459 df-uz 11769 df-fz 12409 df-fzo 12549 df-hash 13201 df-word 13374 df-wlks 26594 df-wlkson 26595 df-trls 26688 df-trlson 26689 df-pths 26711 df-spths 26712 df-pthson 26713 df-spthson 26714 df-wwlksnon 26824 df-wspthsnon 26826 |
This theorem is referenced by: frgrhash2wsp 27375 |
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