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| 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.) | 
| Ref | Expression | 
|---|---|
| elneldisj.e | ⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} | 
| elneldisj.n | ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} | 
| Ref | Expression | 
|---|---|
| elneldisj | ⊢ (𝐸 ∩ 𝑁) = ∅ | 
| 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 3047 | . . . . 5 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
| 4 | 3 | rabbii 3442 | . . . 4 ⊢ {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶} | 
| 5 | 2, 4 | eqtri 2765 | . . 3 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶} | 
| 6 | 1, 5 | ineq12i 4218 | . 2 ⊢ (𝐸 ∩ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}) | 
| 7 | rabnc 4391 | . 2 ⊢ ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}) = ∅ | |
| 8 | 6, 7 | eqtri 2765 | 1 ⊢ (𝐸 ∩ 𝑁) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 {crab 3436 ∩ cin 3950 ∅c0 4333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nel 3047 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-nul 4334 | 
| This theorem is referenced by: cusgrsizeinds 29470 vtxdginducedm1 29561 | 
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