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 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝑠}
elneldisj.f	⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝑠}

Assertion

Ref	Expression
elnelun	⊢ (𝐸 ∪ 𝑁) = 𝐴

Distinct variable group:   𝐴,𝑠

Allowed substitution hints:   𝐵(𝑠)   𝐸(𝑠)   𝑁(𝑠)

Proof of Theorem elnelun

Step	Hyp	Ref	Expression
1		elneldisj.e	. . 3 ⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝑠}
2		elneldisj.f	. . . 4 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝑠}
3		df-nel 2783	. . . . . 6 ⊢ (𝐵 ∉ 𝑠 ↔ ¬ 𝐵 ∈ 𝑠)
4	3	a1i 11	. . . . 5 ⊢ (𝑠 ∈ 𝐴 → (𝐵 ∉ 𝑠 ↔ ¬ 𝐵 ∈ 𝑠))
5	4	rabbiia 3161	. . . 4 ⊢ {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝑠} = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝑠}
6	2, 5	eqtri 2632	. . 3 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝑠}
7	1, 6	uneq12i 3727	. 2 ⊢ (𝐸 ∪ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝑠} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝑠})
8		rabxm 3915	. 2 ⊢ 𝐴 = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝑠} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝑠})
9	7, 8	eqtr4i 2635	1 ⊢ (𝐸 ∪ 𝑁) = 𝐴

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