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Mirrors > Home > MPE Home > Th. List > elneldisj | Structured version Visualization version GIF version |
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 3092 | . . . . 5 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
4 | 3 | rabbii 3420 | . . . 4 ⊢ {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶} |
5 | 2, 4 | eqtri 2821 | . . 3 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶} |
6 | 1, 5 | ineq12i 4137 | . 2 ⊢ (𝐸 ∩ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}) |
7 | rabnc 4295 | . 2 ⊢ ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∩ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}) = ∅ | |
8 | 6, 7 | eqtri 2821 | 1 ⊢ (𝐸 ∩ 𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 {crab 3110 ∩ cin 3880 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-nel 3092 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-nul 4244 |
This theorem is referenced by: cusgrsizeinds 27242 vtxdginducedm1 27333 |
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