Theorem elnelun 3964

Description: The union of 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 yields the original set. (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 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
elneldisj.n	⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶}

Assertion

Ref	Expression
elnelun	⊢ (𝐸 ∪ 𝑁) = 𝐴

Distinct variable group:   𝐴,𝑠

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

Proof of Theorem elnelun

Step	Hyp	Ref	Expression
1		elneldisj.e	. . 3 ⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
2		elneldisj.n	. . . 4 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶}
3		df-nel 2898	. . . . . 6 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
4	3	a1i 11	. . . . 5 ⊢ (𝑠 ∈ 𝐴 → (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶))
5	4	rabbiia 3185	. . . 4 ⊢ {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}
6	2, 5	eqtri 2644	. . 3 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}
7	1, 6	uneq12i 3765	. 2 ⊢ (𝐸 ∪ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶})
8		rabxm 3961	. 2 ⊢ 𝐴 = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶})
9	7, 8	eqtr4i 2647	1 ⊢ (𝐸 ∪ 𝑁) = 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1483   ∈ wcel 1990   ∉ wnel 2897  {crab 2916   ∪ cun 3572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rab 2921  df-v 3202  df-un 3579

This theorem is referenced by:  usgrfilem  26219  cusgrsizeinds  26348  vtxdginducedm1  26439