Theorem elnelun 4346

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 3066	. . . 4 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
4	2, 3	rabbieq 3424	. . 3 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}
5	1, 4	uneq12i 4168	. 2 ⊢ (𝐸 ∪ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶})
6		rabxm 4343	. 2 ⊢ 𝐴 = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶})
7	5, 6	eqtr4i 2789	1 ⊢ (𝐸 ∪ 𝑁) = 𝐴

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147   ∉ wnel 3065  {crab 3416   ∪ cun 4152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nel 3066  df-ral 3081  df-rab 3417  df-v 3456  df-un 4154

This theorem is used by:  usgrfilem  29965  cusgrsizeinds  30090  vtxdginducedm1  30181