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Theorem 2wspiundisj 26718
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.)
Hypothesis
Ref Expression
2wspdisj.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
2wspiundisj Disj 𝑎𝑉 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)
Distinct variable groups:   𝐺,𝑏   𝑉,𝑏   𝐺,𝑎,𝑏   𝑉,𝑎

Proof of Theorem 2wspiundisj
Dummy variables 𝑐 𝑡 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 400 . . . . 5 (𝑎 = 𝑐 → (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅))
21a1d 25 . . . 4 (𝑎 = 𝑐 → ((𝑎𝑉𝑐𝑉) → (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅)))
3 eliun 4495 . . . . . . . . . 10 (𝑡 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ↔ ∃𝑏 ∈ (𝑉 ∖ {𝑎})𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏))
4 eliun 4495 . . . . . . . . . . . . . . 15 (𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑) ↔ ∃𝑑 ∈ (𝑉 ∖ {𝑐})𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑))
5 wspthneq1eq2 26608 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑)) → (𝑎 = 𝑐𝑏 = 𝑑))
65simpld 475 . . . . . . . . . . . . . . . . . 18 ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑)) → 𝑎 = 𝑐)
76ex 450 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → (𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
87adantr 481 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → (𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
98rexlimdva 3029 . . . . . . . . . . . . . . 15 (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → (∃𝑑 ∈ (𝑉 ∖ {𝑐})𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
104, 9syl5bi 232 . . . . . . . . . . . . . 14 (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → (𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
1110con3rr3 151 . . . . . . . . . . . . 13 𝑎 = 𝑐 → (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
1211adantr 481 . . . . . . . . . . . 12 ((¬ 𝑎 = 𝑐 ∧ (𝑎𝑉𝑐𝑉)) → (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
1312adantr 481 . . . . . . . . . . 11 (((¬ 𝑎 = 𝑐 ∧ (𝑎𝑉𝑐𝑉)) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
1413rexlimdva 3029 . . . . . . . . . 10 ((¬ 𝑎 = 𝑐 ∧ (𝑎𝑉𝑐𝑉)) → (∃𝑏 ∈ (𝑉 ∖ {𝑎})𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
153, 14syl5bi 232 . . . . . . . . 9 ((¬ 𝑎 = 𝑐 ∧ (𝑎𝑉𝑐𝑉)) → (𝑡 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
1615ralrimiv 2964 . . . . . . . 8 ((¬ 𝑎 = 𝑐 ∧ (𝑎𝑉𝑐𝑉)) → ∀𝑡 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑))
17 oveq2 6613 . . . . . . . . . . . 12 (𝑏 = 𝑑 → (𝑐(2 WSPathsNOn 𝐺)𝑏) = (𝑐(2 WSPathsNOn 𝐺)𝑑))
1817cbviunv 4530 . . . . . . . . . . 11 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) = 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)
1918eleq2i 2696 . . . . . . . . . 10 (𝑡 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) ↔ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑))
2019notbii 310 . . . . . . . . 9 𝑡 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) ↔ ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑))
2120ralbii 2979 . . . . . . . 8 (∀𝑡 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ¬ 𝑡 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) ↔ ∀𝑡 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ¬ 𝑡 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑))
2216, 21sylibr 224 . . . . . . 7 ((¬ 𝑎 = 𝑐 ∧ (𝑎𝑉𝑐𝑉)) → ∀𝑡 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ¬ 𝑡 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏))
23 disj 3994 . . . . . . 7 (( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅ ↔ ∀𝑡 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ¬ 𝑡 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏))
2422, 23sylibr 224 . . . . . 6 ((¬ 𝑎 = 𝑐 ∧ (𝑎𝑉𝑐𝑉)) → ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅)
2524olcd 408 . . . . 5 ((¬ 𝑎 = 𝑐 ∧ (𝑎𝑉𝑐𝑉)) → (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅))
2625ex 450 . . . 4 𝑎 = 𝑐 → ((𝑎𝑉𝑐𝑉) → (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅)))
272, 26pm2.61i 176 . . 3 ((𝑎𝑉𝑐𝑉) → (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅))
2827rgen2 2974 . 2 𝑎𝑉𝑐𝑉 (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅)
29 sneq 4163 . . . . 5 (𝑎 = 𝑐 → {𝑎} = {𝑐})
3029difeq2d 3711 . . . 4 (𝑎 = 𝑐 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑐}))
31 oveq1 6612 . . . 4 (𝑎 = 𝑐 → (𝑎(2 WSPathsNOn 𝐺)𝑏) = (𝑐(2 WSPathsNOn 𝐺)𝑏))
3230, 31iuneq12d 4517 . . 3 (𝑎 = 𝑐 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) = 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏))
3332disjor 4602 . 2 (Disj 𝑎𝑉 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ↔ ∀𝑎𝑉𝑐𝑉 (𝑎 = 𝑐 ∨ ( 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅))
3428, 33mpbir 221 1 Disj 𝑎𝑉 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  cdif 3557  cin 3559  c0 3896  {csn 4153   ciun 4490  Disj wdisj 4588  cfv 5850  (class class class)co 6605  2c2 11015  Vtxcvtx 25769   WSPathsNOn cwwspthsnon 26584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-wlks 26359  df-wlkson 26360  df-trls 26452  df-trlson 26453  df-pths 26475  df-spths 26476  df-pthson 26477  df-spthson 26478  df-wwlksnon 26587  df-wspthsnon 26589
This theorem is referenced by:  frgrhash2wsp  27049
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