Theorem elneldisj 4344

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 3126	. . . . 5 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)
4	3	rabbii 3475	. . . 4 ⊢ {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}
5	2, 4	eqtri 2846	. . 3 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}
6	1, 5	ineq12i 4189	. 2 ⊢ (𝐸 ∩ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶})
7		rabnc 4343	. 2 ⊢ ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}) = ∅
8	6, 7	eqtri 2846	1 ⊢ (𝐸 ∩ 𝑁) = ∅

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114   ∉ wnel 3125  {crab 3144   ∩ cin 3937  ∅c0 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-nel 3126  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-in 3945  df-nul 4294

This theorem is referenced by:  cusgrsizeinds  27236  vtxdginducedm1  27327