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Mirrors > Home > MPE Home > Th. List > elnelun | Structured version Visualization version GIF version |
Description: The union of 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 yields the original set. (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 |
---|---|
elnelun | ⊢ (𝐸 ∪ 𝑁) = 𝐴 |
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 3044 | . . . . 5 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
4 | 3 | rabbii 3435 | . . . 4 ⊢ {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶} |
5 | 2, 4 | eqtri 2756 | . . 3 ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶} |
6 | 1, 5 | uneq12i 4160 | . 2 ⊢ (𝐸 ∪ 𝑁) = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}) |
7 | rabxm 4387 | . 2 ⊢ 𝐴 = ({𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} ∪ {𝑠 ∈ 𝐴 ∣ ¬ 𝐵 ∈ 𝐶}) | |
8 | 6, 7 | eqtr4i 2759 | 1 ⊢ (𝐸 ∪ 𝑁) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 ∉ wnel 3043 {crab 3429 ∪ cun 3945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-nel 3044 df-ral 3059 df-rab 3430 df-v 3473 df-un 3952 |
This theorem is referenced by: usgrfilem 29139 cusgrsizeinds 29265 vtxdginducedm1 29356 |
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