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Theorem clwwlksndisj 26852
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 3880 . . . . 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 2641 . . . . . . . 8 (((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦) → 𝑥 = 𝑦)
32con3i 150 . . . . . . 7 𝑥 = 𝑦 → ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
43ralrimivw 2962 . . . . . 6 𝑥 = 𝑦 → ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
5 rabeq0 3936 . . . . . 6 ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅ ↔ ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
64, 5sylibr 224 . . . . 5 𝑥 = 𝑦 → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅)
71, 6syl5eq 2667 . . . 4 𝑥 = 𝑦 → ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
87orri 391 . . 3 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
98rgen2w 2920 . 2 𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
10 eqeq2 2632 . . . 4 (𝑥 = 𝑦 → ((𝑤‘0) = 𝑥 ↔ (𝑤‘0) = 𝑦))
1110rabbidv 3180 . . 3 (𝑥 = 𝑦 → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦})
1211disjor 4602 . 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 221 1 Disj 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 383  wa 384   = wceq 1480  wral 2907  {crab 2911  cin 3558  c0 3896  Disj wdisj 4588  cfv 5852  (class class class)co 6610  0cc0 9887   ClWWalksN cclwwlksn 26756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rmo 2915  df-rab 2916  df-v 3191  df-dif 3562  df-in 3566  df-nul 3897  df-disj 4589
This theorem is referenced by:  numclwwlk4  27111
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