Theorem clwwlksndisj 26973

Description: The sets of closed walks starting at different vertices are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.)

Assertion

Ref	Expression
clwwlksndisj	⊢ Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥}

Distinct variable groups:   𝑥,𝐺   𝑥,𝑁   𝑥,𝑉   𝑥,𝑤

Allowed substitution hints:   𝐺(𝑤)   𝑁(𝑤)   𝑉(𝑤)

Proof of Theorem clwwlksndisj

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inrab 3899	. . . . 5 ⊢ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)}
2		eqtr2 2642	. . . . . . . 8 ⊢ (((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦) → 𝑥 = 𝑦)
3	2	con3i 150	. . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 → ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
4	3	ralrimivw 2967	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
5		rabeq0 3957	. . . . . 6 ⊢ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅ ↔ ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
6	4, 5	sylibr 224	. . . . 5 ⊢ (¬ 𝑥 = 𝑦 → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅)
7	1, 6	syl5eq 2668	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
8	7	orri 391	. . 3 ⊢ (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
9	8	rgen2w 2925	. 2 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
10		eqeq2 2633	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑤‘0) = 𝑥 ↔ (𝑤‘0) = 𝑦))
11	10	rabbidv 3189	. . 3 ⊢ (𝑥 = 𝑦 → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦})
12	11	disjor 4634	. 2 ⊢ (Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅))
13	9, 12	mpbir 221	1 ⊢ Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥}

Syntax hints:  ¬ wn 3   ∨ wo 383   ∧ wa 384   = wceq 1483  ∀wral 2912  {crab 2916   ∩ cin 3573  ∅c0 3915  Disj wdisj 4620  ‘cfv 5888  (class class class)co 6650  0cc0 9936   ClWWalksN cclwwlksn 26876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by:  numclwwlk4  27244