Theorem disjxwwlksnOLD 27293

Description: Obsolete version of disjxwwlksn 27292 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 29-Jul-2018.) (Revised by AV, 19-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
wwlksnexthasheq.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnexthasheq.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
disjxwwlksnOLD	⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}

Distinct variable groups:   𝑦,𝑁   𝑥,𝑉   𝑥,𝑦

Allowed substitution hints:   𝑃(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑁(𝑥)   𝑉(𝑦)

Proof of Theorem disjxwwlksnOLD

Step	Hyp	Ref	Expression
1		simp1 1127	. . . . 5 ⊢ (((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 substr ⟨0, 𝑁⟩) = 𝑦)
2	1	a1i 11	. . . 4 ⊢ (𝑥 ∈ Word 𝑉 → (((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 substr ⟨0, 𝑁⟩) = 𝑦))
3	2	ss2rabi 3905	. . 3 ⊢ {𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}
4	3	rgenw 3106	. 2 ⊢ ∀𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}
5		disjxwrdOLD 13781	. 2 ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}
6		disjss2 4859	. 2 ⊢ (∀𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦} → (Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦} → Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}))
7	4, 5, 6	mp2 9	1 ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}

Syntax hints:   → wi 4   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  {crab 3094   ⊆ wss 3792  {cpr 4400  ⟨cop 4404  Disj wdisj 4856  ‘cfv 6137  (class class class)co 6924  0cc0 10274  Word cword 13605  lastSclsw 13658   substr csubstr 13736  Vtxcvtx 26361  Edgcedg 26412   WWalksN cwwlksn 27192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-in 3799  df-ss 3806  df-disj 4857

This theorem is referenced by: (None)