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Mirrors > Home > ILE Home > Th. List > notrab | GIF version |
Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
notrab | ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difab 3386 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)} | |
2 | difin 3354 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) = (𝐴 ∖ {𝑥 ∣ 𝜑}) | |
3 | dfrab3 3393 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
4 | 3 | difeq2i 3232 | . . 3 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) |
5 | abid2 2285 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
6 | 5 | difeq1i 3231 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = (𝐴 ∖ {𝑥 ∣ 𝜑}) |
7 | 2, 4, 6 | 3eqtr4i 2195 | . 2 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) |
8 | df-rab 2451 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)} | |
9 | 1, 7, 8 | 3eqtr4i 2195 | 1 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1342 ∈ wcel 2135 {cab 2150 {crab 2446 ∖ cdif 3108 ∩ cin 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rab 2451 df-v 2723 df-dif 3113 df-in 3117 |
This theorem is referenced by: diffitest 6844 |
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