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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 3345 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)} | |
2 | difin 3313 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) = (𝐴 ∖ {𝑥 ∣ 𝜑}) | |
3 | dfrab3 3352 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
4 | 3 | difeq2i 3191 | . . 3 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥 ∣ 𝜑})) |
5 | abid2 2260 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
6 | 5 | difeq1i 3190 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) = (𝐴 ∖ {𝑥 ∣ 𝜑}) |
7 | 2, 4, 6 | 3eqtr4i 2170 | . 2 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = ({𝑥 ∣ 𝑥 ∈ 𝐴} ∖ {𝑥 ∣ 𝜑}) |
8 | df-rab 2425 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)} | |
9 | 1, 7, 8 | 3eqtr4i 2170 | 1 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {cab 2125 {crab 2420 ∖ cdif 3068 ∩ cin 3070 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rab 2425 df-v 2688 df-dif 3073 df-in 3077 |
This theorem is referenced by: diffitest 6781 |
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