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Theorem elnelun 3943
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 2894 . . . . . 6 (𝐵𝑠 ↔ ¬ 𝐵𝑠)
43a1i 11 . . . . 5 (𝑠𝐴 → (𝐵𝑠 ↔ ¬ 𝐵𝑠))
54rabbiia 3176 . . . 4 {𝑠𝐴𝐵𝑠} = {𝑠𝐴 ∣ ¬ 𝐵𝑠}
62, 5eqtri 2643 . . 3 𝑁 = {𝑠𝐴 ∣ ¬ 𝐵𝑠}
71, 6uneq12i 3748 . 2 (𝐸𝑁) = ({𝑠𝐴𝐵𝑠} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝑠})
8 rabxm 3940 . 2 𝐴 = ({𝑠𝐴𝐵𝑠} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝑠})
97, 8eqtr4i 2646 1 (𝐸𝑁) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1480  wcel 1987  wnel 2893  {crab 2911  cun 3557
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-nel 2894  df-ral 2912  df-rab 2916  df-v 3191  df-un 3564
This theorem is referenced by:  usgrfilem  26120  cusgrsizeinds  26248
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