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Theorem clwwlknondisjOLD 27292
Description: Obsolete version of clwwlknondisj 27287 as of 3-Mar-2022. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
clwwlknondisjOLD 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 clwwlknondisjOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 inrab 4047 . . . . 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 2791 . . . . . . . 8 (((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦) → 𝑥 = 𝑦)
32con3i 151 . . . . . . 7 𝑥 = 𝑦 → ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
43ralrimivw 3116 . . . . . 6 𝑥 = 𝑦 → ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
5 rabeq0 4103 . . . . . 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 2817 . . . 4 𝑥 = 𝑦 → ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
87orri 849 . . 3 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
98rgen2w 3074 . 2 𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
10 eqeq2 2782 . . . 4 (𝑥 = 𝑦 → ((𝑤‘0) = 𝑥 ↔ (𝑤‘0) = 𝑦))
1110rabbidv 3339 . . 3 (𝑥 = 𝑦 → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦})
1211disjor 4768 . 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  wa 382  wo 834   = wceq 1631  wral 3061  {crab 3065  cin 3722  c0 4063  Disj wdisj 4754  cfv 6031  (class class class)co 6793  0cc0 10138   ClWWalksN cclwwlkn 27174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rmo 3069  df-rab 3070  df-v 3353  df-dif 3726  df-in 3730  df-nul 4064  df-disj 4755
This theorem is referenced by: (None)
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