Theorem elwspths2onw 30564

Description: A simple path of length 2 between two vertices (in a simple pseudograph) as length 3 string. This theorem avoids the Axiom of Choice for its proof, at the cost of requiring a simple graph; the more general version is elwspths2on 30563. (Contributed by Ender Ting, 29-Jan-2026.)

Hypothesis

Ref	Expression
elwwlks2on.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
elwspths2onw	⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐺,𝑏   𝑉,𝑏   𝑊,𝑏

Proof of Theorem elwspths2onw

Step	Hyp	Ref	Expression
1		wspthnon 30459	. . . 4 ⊢ (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ ∃𝑏 𝑏(𝐴(SPathsOn‘𝐺)𝐶)𝑊))
2	1	biimpi 219	. . 3 ⊢ (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ ∃𝑏 𝑏(𝐴(SPathsOn‘𝐺)𝐶)𝑊))
3		elwwlks2on.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	elwwlks2ons3 30556	. . . . . . 7 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
5	4	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
6		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝑊 = ⟨“𝐴𝑏𝐶”⟩)
7		eleq1 2851	. . . . . . . . . . . . . 14 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
8	7	biimpa 482	. . . . . . . . . . . . 13 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))
9	6, 8	jca 521	. . . . . . . . . . . 12 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
10	9	ex 418	. . . . . . . . . . 11 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
11	10	adantr 486	. . . . . . . . . 10 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
12	11	com12 33	. . . . . . . . 9 ⊢ (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
13	12	reximdv 3180	. . . . . . . 8 ⊢ (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
14	13	a1i13 28	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (∃𝑏 𝑏(𝐴(SPathsOn‘𝐺)𝐶)𝑊 → (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))))
15	14	com24 96	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (∃𝑏 𝑏(𝐴(SPathsOn‘𝐺)𝐶)𝑊 → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))))
16	5, 15	sylbid 243	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (∃𝑏 𝑏(𝐴(SPathsOn‘𝐺)𝐶)𝑊 → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))))
17	16	impd 416	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ ∃𝑏 𝑏(𝐴(SPathsOn‘𝐺)𝐶)𝑊) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))))
18	17	com23 87	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ ∃𝑏 𝑏(𝐴(SPathsOn‘𝐺)𝐶)𝑊) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))))
19	2, 18	mpdi 46	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
20	7	biimpar 483	. . . 4 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))
21	20	a1i 11	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
22	21	rexlimdva 3166	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
23	19, 22	impbid 215	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2146  ∃wrex 3089   class class class wbr 5105  ‘cfv 6546  (class class class)co 7427  2c2 12407  ⟨“cs3 15004  Vtxcvtx 29586  USPGraphcuspgr 29741  SPathsOncspthson 30310   WWalksNOn cwwlksnon 30428   WSPathsNOn cwwspthsnon 30430

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-er 8728  df-map 8860  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 10030  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-3 12416  df-n0 12617  df-z 12704  df-uz 12976  df-fz 13650  df-fzo 13799  df-hash 14485  df-word 14669  df-concat 14726  df-s1 14753  df-s2 15010  df-s3 15011  df-wwlks 30431  df-wwlksn 30432  df-wwlksnon 30433  df-wspthsnon 30435

This theorem is used by:  usgr2wspthon  30569