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Theorem clwwlksndisj 41280
 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 𝑥𝑉 {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.
StepHypRef Expression
1 inrab 3858 . . . . 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 2630 . . . . . . . 8 (((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦) → 𝑥 = 𝑦)
32con3i 149 . . . . . . 7 𝑥 = 𝑦 → ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
43ralrimivw 2950 . . . . . 6 𝑥 = 𝑦 → ∀𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
5 rabeq0 3911 . . . . . 6 ({𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅ ↔ ∀𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
64, 5sylibr 223 . . . . 5 𝑥 = 𝑦 → {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅)
71, 6syl5eq 2656 . . . 4 𝑥 = 𝑦 → ({𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
87orri 390 . . 3 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
98rgen2w 2909 . 2 𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
10 eqeq2 2621 . . . 4 (𝑥 = 𝑦 → ((𝑤‘0) = 𝑥 ↔ (𝑤‘0) = 𝑦))
1110rabbidv 3164 . . 3 (𝑥 = 𝑦 → {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} = {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑦})
1211disjor 4567 . 2 (Disj 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ ∀𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅))
139, 12mpbir 220 1 Disj 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑥}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383   = wceq 1475  ∀wral 2896  {crab 2900   ∩ cin 3539  ∅c0 3874  Disj wdisj 4553  ‘cfv 5804  (class class class)co 6549  0cc0 9815   ClWWalkSN cclwwlksn 41184 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:  av-numclwwlk4  41540
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