Theorem elneldisj 4320

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 3039	. . . 4 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
4	2, 3	rabbieq 3399	. . 3 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}
5	1, 4	ineq12i 4147	. 2 ⊢ (𝐸 ∩ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶})
6		rabnc 4319	. 2 ⊢ ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}) = ∅
7	5, 6	eqtri 2762	1 ⊢ (𝐸 ∩ 𝑁) = ∅

Syntax hints:  ¬ wn 3   = wceq 1547   ∈ wcel 2119   ∉ wnel 3038  {crab 3391   ∩ cin 3882  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nel 3039  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-in 3890  df-nul 4262

This theorem is referenced by:  cusgrsizeinds  29539  vtxdginducedm1  29630