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Theorem elneldisjOLD 4129
Description: Obsolete version of elneldisj 4127 as of 17-Dec-2021. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elneldisjOLD.e 𝐸 = {𝑠𝐴𝐵𝑠}
elneldisjOLD.f 𝑁 = {𝑠𝐴𝐵𝑠}
Assertion
Ref Expression
elneldisjOLD (𝐸𝑁) = ∅
Distinct variable group:   𝐴,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝐸(𝑠)   𝑁(𝑠)

Proof of Theorem elneldisjOLD
StepHypRef Expression
1 elneldisjOLD.e . . 3 𝐸 = {𝑠𝐴𝐵𝑠}
2 elneldisjOLD.f . . . 4 𝑁 = {𝑠𝐴𝐵𝑠}
3 df-nel 3041 . . . . . 6 (𝐵𝑠 ↔ ¬ 𝐵𝑠)
43a1i 11 . . . . 5 (𝑠𝐴 → (𝐵𝑠 ↔ ¬ 𝐵𝑠))
54rabbiia 3333 . . . 4 {𝑠𝐴𝐵𝑠} = {𝑠𝐴 ∣ ¬ 𝐵𝑠}
62, 5eqtri 2787 . . 3 𝑁 = {𝑠𝐴 ∣ ¬ 𝐵𝑠}
71, 6ineq12i 3976 . 2 (𝐸𝑁) = ({𝑠𝐴𝐵𝑠} ∩ {𝑠𝐴 ∣ ¬ 𝐵𝑠})
8 rabnc 4126 . 2 ({𝑠𝐴𝐵𝑠} ∩ {𝑠𝐴 ∣ ¬ 𝐵𝑠}) = ∅
97, 8eqtri 2787 1 (𝐸𝑁) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197   = wceq 1652  wcel 2155  wnel 3040  {crab 3059  cin 3733  c0 4081
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 3737  df-in 3741  df-nul 4082
This theorem is referenced by: (None)
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