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 𝑎 ∈ 𝑉 ∪ 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.

Step	Hyp	Ref	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)) = ∅))
2	1	a1d 25	. . . 4 ⊢ (𝑎 = 𝑐 → ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (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 = 𝑐 ∧ 𝑏 = 𝑑))
6	5	simpld 475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑)) → 𝑎 = 𝑐)
7	6	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → (𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
8	7	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → (𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
9	8	rexlimdva 3029	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → (∃𝑑 ∈ (𝑉 ∖ {𝑐})𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
10	4, 9	syl5bi 232	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → (𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
11	10	con3rr3 151	. . . . . . . . . . . . 13 ⊢ (¬ 𝑎 = 𝑐 → (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
12	11	adantr 481	. . . . . . . . . . . 12 ⊢ ((¬ 𝑎 = 𝑐 ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
13	12	adantr 481	. . . . . . . . . . 11 ⊢ (((¬ 𝑎 = 𝑐 ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
14	13	rexlimdva 3029	. . . . . . . . . 10 ⊢ ((¬ 𝑎 = 𝑐 ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (∃𝑏 ∈ (𝑉 ∖ {𝑎})𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
15	3, 14	syl5bi 232	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑐 ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
16	15	ralrimiv 2964	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑐 ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ∀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))
18	17	cbviunv 4530	. . . . . . . . . . 11 ⊢ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) = ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)
19	18	eleq2i 2696	. . . . . . . . . 10 ⊢ (𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) ↔ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑))
20	19	notbii 310	. . . . . . . . 9 ⊢ (¬ 𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) ↔ ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑))
21	20	ralbii 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))
22	16, 21	sylibr 224	. . . . . . 7 ⊢ ((¬ 𝑎 = 𝑐 ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ∀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))
24	22, 23	sylibr 224	. . . . . 6 ⊢ ((¬ 𝑎 = 𝑐 ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅)
25	24	olcd 408	. . . . 5 ⊢ ((¬ 𝑎 = 𝑐 ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅))
26	25	ex 450	. . . 4 ⊢ (¬ 𝑎 = 𝑐 → ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅)))
27	2, 26	pm2.61i 176	. . 3 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅))
28	27	rgen2 2974	. 2 ⊢ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ 𝑉 (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})
30	29	difeq2d 3711	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑐}))
31		oveq1 6612	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑎(2 WSPathsNOn 𝐺)𝑏) = (𝑐(2 WSPathsNOn 𝐺)𝑏))
32	30, 31	iuneq12d 4517	. . 3 ⊢ (𝑎 = 𝑐 → ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) = ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏))
33	32	disjor 4602	. 2 ⊢ (Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ↔ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ 𝑉 (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ∩ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏)) = ∅))
34	28, 33	mpbir 221	1 ⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)

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