Theorem elneldisj 4345

Description: 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 are disjoint. (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
elneldisj	⊢ (𝐸 ∩ 𝑁) = ∅

Distinct variable group:   𝐴,𝑠

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

Proof of Theorem elneldisj

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	ineq12i 4118	. 2 ⊢ (𝐸 ∩ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶})
6		rabnc 4344	. 2 ⊢ ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}) = ∅
7	5, 6	eqtri 2786	1 ⊢ (𝐸 ∩ 𝑁) = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147   ∉ wnel 3065  {crab 3416   ∩ cin 4092  ∅c0 4281

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-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  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-dif 4050  df-in 4094  df-nul 4282

This theorem is used by:  cusgrsizeinds  30090  vtxdginducedm1  30181