Theorem numclwwlkdisj 26607

Description: The sets of closed walks starting at different vertices in an undirected simple graph are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.)

Hypothesis

Ref	Expression
numclwwlk.c	⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))

Assertion

Ref	Expression
numclwwlkdisj	⊢ Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥}

Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑤,𝐶,𝑥   𝑥,𝐸   𝑤,𝑁,𝑥   𝑥,𝑉

Allowed substitution hints:   𝐶(𝑛)   𝐸(𝑤)   𝑉(𝑤)

Proof of Theorem numclwwlkdisj

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inrab 3858	. . . . 5 ⊢ ({𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑦}) = {𝑤 ∈ (𝐶‘𝑁) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)}
2		eqtr2 2630	. . . . . . . 8 ⊢ (((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦) → 𝑥 = 𝑦)
3	2	con3i 149	. . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 → ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
4	3	ralrimivw 2950	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → ∀𝑤 ∈ (𝐶‘𝑁) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
5		rabeq0 3911	. . . . . 6 ⊢ ({𝑤 ∈ (𝐶‘𝑁) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅ ↔ ∀𝑤 ∈ (𝐶‘𝑁) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
6	4, 5	sylibr 223	. . . . 5 ⊢ (¬ 𝑥 = 𝑦 → {𝑤 ∈ (𝐶‘𝑁) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅)
7	1, 6	syl5eq 2656	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → ({𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑦}) = ∅)
8	7	orri 390	. . 3 ⊢ (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑦}) = ∅)
9	8	rgen2w 2909	. 2 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑦}) = ∅)
10		eqeq2 2621	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑤‘0) = 𝑥 ↔ (𝑤‘0) = 𝑦))
11	10	rabbidv 3164	. . 3 ⊢ (𝑥 = 𝑦 → {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥} = {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑦})
12	11	disjor 4567	. 2 ⊢ (Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥} ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑦}) = ∅))
13	9, 12	mpbir 220	1 ⊢ Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥}

Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383   = wceq 1475  ∀wral 2896  {crab 2900   ∩ cin 3539  ∅c0 3874  Disj wdisj 4553   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ₀cn0 11169   ClWWalksN cclwwlkn 26277

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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rmo 2904  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-nul 3875  df-disj 4554

This theorem is referenced by:  numclwwlk4  26637