Theorem usg2spthonot1 26417

Description: A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.)

Assertion

Ref	Expression
usg2spthonot1	⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐸,𝑏   𝑇,𝑏   𝑉,𝑏

Proof of Theorem usg2spthonot1

Step	Hyp	Ref	Expression
1		usgrav 25867	. . 3 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2		el2spthonot0 26398	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))
3	1, 2	sylan 487	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))
4		simpll 786	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝑉 USGrph 𝐸)
5		simplrl 796	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐴 ∈ 𝑉)
6		simpr 476	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
7		simplrr 797	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → 𝐶 ∈ 𝑉)
8		usg2spthonot0 26416	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ((𝐴 = 𝐴 ∧ 𝐶 = 𝐶 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))
9	4, 5, 6, 7, 8	syl13anc 1320	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ((𝐴 = 𝐴 ∧ 𝐶 = 𝐶 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))
10		simp3 1056	. . . . . . . 8 ⊢ ((𝐴 = 𝐴 ∧ 𝐶 = 𝐶 ∧ 𝐴 ≠ 𝐶) → 𝐴 ≠ 𝐶)
11	10	anim1i 590	. . . . . . 7 ⊢ (((𝐴 = 𝐴 ∧ 𝐶 = 𝐶 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)) → (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
12		eqidd 2611	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐶 → 𝐴 = 𝐴)
13		eqidd 2611	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐶 → 𝐶 = 𝐶)
14		id 22	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐶 → 𝐴 ≠ 𝐶)
15	12, 13, 14	3jca 1235	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐶 → (𝐴 = 𝐴 ∧ 𝐶 = 𝐶 ∧ 𝐴 ≠ 𝐶))
16	15	anim1i 590	. . . . . . 7 ⊢ ((𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)) → ((𝐴 = 𝐴 ∧ 𝐶 = 𝐶 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
17	11, 16	impbii 198	. . . . . 6 ⊢ (((𝐴 = 𝐴 ∧ 𝐶 = 𝐶 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)) ↔ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
18	9, 17	syl6bb 275	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))
19	18	anbi2d 736	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)) ↔ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))))
20		anass 679	. . . 4 ⊢ (((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)) ↔ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴 ≠ 𝐶 ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))
21	19, 20	syl6bbr 277	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)) ↔ ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))
22	21	rexbidva 3031	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (∃𝑏 ∈ 𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))
23	3, 22	bitrd 267	1 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173  {cpr 4127  ⟨cotp 4133   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   2SPathOnOt c2pthonot 26384

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-3or 1032  df-3an 1033  df-tru 1478  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-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  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-cda 8873  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-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2wlkonot 26385  df-2spthonot 26387

This theorem is referenced by:  usg2spot2nb  26592