Theorem elnelun 4416

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 3053	. . . . 5 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
4	3	rabbii 3449	. . . 4 ⊢ {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}
5	2, 4	eqtri 2768	. . 3 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}
6	1, 5	uneq12i 4189	. 2 ⊢ (𝐸 ∪ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶})
7		rabxm 4413	. 2 ⊢ 𝐴 = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶})
8	6, 7	eqtr4i 2771	1 ⊢ (𝐸 ∪ 𝑁) = 𝐴

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108   ∉ wnel 3052  {crab 3443   ∪ cun 3974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nel 3053  df-ral 3068  df-rab 3444  df-v 3490  df-un 3981

This theorem is referenced by:  usgrfilem  29364  cusgrsizeinds  29490  vtxdginducedm1  29581