Theorem 2wspiundisj 41166

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 399	. . . . 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 4460	. . . . . . . . . 10 ⊢ (𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) ↔ ∃𝑏 ∈ (𝑉 ∖ {𝑎})𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏))
4		eliun 4460	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑) ↔ ∃𝑑 ∈ (𝑉 ∖ {𝑐})𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑))
5		wspthneq1eq2 41056	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑)) → (𝑎 = 𝑐 ∧ 𝑏 = 𝑑))
6	5	simpld 474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑)) → 𝑎 = 𝑐)
7	6	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → (𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
8	7	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) ∧ 𝑑 ∈ (𝑉 ∖ {𝑐})) → (𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
9	8	rexlimdva 3013	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → (∃𝑑 ∈ (𝑉 ∖ {𝑐})𝑡 ∈ (𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
10	4, 9	syl5bi 231	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → (𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑) → 𝑎 = 𝑐))
11	10	con3rr3 150	. . . . . . . . . . . . 13 ⊢ (¬ 𝑎 = 𝑐 → (𝑡 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑏) → ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)))
12	11	adantr 480	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 3013	. . . . . . . . . 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 231	. . . . . . . . 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 2948	. . . . . . . 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 6557	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → (𝑐(2 WSPathsNOn 𝐺)𝑏) = (𝑐(2 WSPathsNOn 𝐺)𝑑))
18	17	cbviunv 4495	. . . . . . . . . . 11 ⊢ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) = ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑)
19	18	eleq2i 2680	. . . . . . . . . 10 ⊢ (𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) ↔ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑))
20	19	notbii 309	. . . . . . . . 9 ⊢ (¬ 𝑡 ∈ ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏) ↔ ¬ 𝑡 ∈ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑑))
21	20	ralbii 2963	. . . . . . . 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 223	. . . . . . 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 3969	. . . . . . 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 223	. . . . . 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 407	. . . . 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 449	. . . 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 175	. . 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 2958	. 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 4135	. . . . 5 ⊢ (𝑎 = 𝑐 → {𝑎} = {𝑐})
30	29	difeq2d 3690	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑐}))
31		oveq1 6556	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑎(2 WSPathsNOn 𝐺)𝑏) = (𝑐(2 WSPathsNOn 𝐺)𝑏))
32	30, 31	iuneq12d 4482	. . 3 ⊢ (𝑎 = 𝑐 → ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) = ∪ 𝑏 ∈ (𝑉 ∖ {𝑐})(𝑐(2 WSPathsNOn 𝐺)𝑏))
33	32	disjor 4567	. 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 220	1 ⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539  ∅c0 3874  {csn 4125  ∪ ciun 4455  Disj wdisj 4553  ‘cfv 5804  (class class class)co 6549  2c2 10947  Vtxcvtx 25673   WSPathsNOn cwwspthsnon 41032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-1wlks 40800  df-wlkson 40802  df-trls 40901  df-trlson 40902  df-pths 40923  df-spths 40924  df-pthson 40925  df-spthson 40926  df-wwlksnon 41035  df-wspthsnon 41037

This theorem is referenced by:  frgrhash2wsp  41497