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Theorem elneldisj 4125
Description: The set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element are disjoint. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
Hypotheses
Ref Expression
elneldisj.e 𝐸 = {𝑠𝐴𝐵𝐶}
elneldisj.n 𝑁 = {𝑠𝐴𝐵𝐶}
Assertion
Ref Expression
elneldisj (𝐸𝑁) = ∅
Distinct variable group:   𝐴,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝐶(𝑠)   𝐸(𝑠)   𝑁(𝑠)

Proof of Theorem elneldisj
StepHypRef Expression
1 elneldisj.e . . 3 𝐸 = {𝑠𝐴𝐵𝐶}
2 elneldisj.n . . . 4 𝑁 = {𝑠𝐴𝐵𝐶}
3 df-nel 3041 . . . . 5 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
43rabbii 3334 . . . 4 {𝑠𝐴𝐵𝐶} = {𝑠𝐴 ∣ ¬ 𝐵𝐶}
52, 4eqtri 2787 . . 3 𝑁 = {𝑠𝐴 ∣ ¬ 𝐵𝐶}
61, 5ineq12i 3974 . 2 (𝐸𝑁) = ({𝑠𝐴𝐵𝐶} ∩ {𝑠𝐴 ∣ ¬ 𝐵𝐶})
7 rabnc 4124 . 2 ({𝑠𝐴𝐵𝐶} ∩ {𝑠𝐴 ∣ ¬ 𝐵𝐶}) = ∅
86, 7eqtri 2787 1 (𝐸𝑁) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1652  wcel 2155  wnel 3040  {crab 3059  cin 3731  c0 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-nel 3041  df-ral 3060  df-rab 3064  df-v 3352  df-dif 3735  df-in 3739  df-nul 4080
This theorem is referenced by:  cusgrsizeinds  26639  vtxdginducedm1  26730
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