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Theorem elnelun 3918
 Description: The union of the set of elements containing a special element and of the set of elements not containing the special element yields the original set. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
Hypotheses
Ref Expression
elneldisj.e 𝐸 = {𝑠𝐴𝐵𝑠}
elneldisj.f 𝑁 = {𝑠𝐴𝐵𝑠}
Assertion
Ref Expression
elnelun (𝐸𝑁) = 𝐴
Distinct variable group:   𝐴,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝐸(𝑠)   𝑁(𝑠)

Proof of Theorem elnelun
StepHypRef Expression
1 elneldisj.e . . 3 𝐸 = {𝑠𝐴𝐵𝑠}
2 elneldisj.f . . . 4 𝑁 = {𝑠𝐴𝐵𝑠}
3 df-nel 2783 . . . . . 6 (𝐵𝑠 ↔ ¬ 𝐵𝑠)
43a1i 11 . . . . 5 (𝑠𝐴 → (𝐵𝑠 ↔ ¬ 𝐵𝑠))
54rabbiia 3161 . . . 4 {𝑠𝐴𝐵𝑠} = {𝑠𝐴 ∣ ¬ 𝐵𝑠}
62, 5eqtri 2632 . . 3 𝑁 = {𝑠𝐴 ∣ ¬ 𝐵𝑠}
71, 6uneq12i 3727 . 2 (𝐸𝑁) = ({𝑠𝐴𝐵𝑠} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝑠})
8 rabxm 3915 . 2 𝐴 = ({𝑠𝐴𝐵𝑠} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝑠})
97, 8eqtr4i 2635 1 (𝐸𝑁) = 𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900   ∪ cun 3538 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rab 2905  df-v 3175  df-un 3545 This theorem is referenced by:  usgrfilem  40546  cusgrsizeinds  40668
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