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Mirrors > Home > ILE Home > Th. List > notab | GIF version |
Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.) |
Ref | Expression |
---|---|
notab | ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2397 | . . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} | |
2 | rabab 2676 | . . 3 ⊢ {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑} | |
3 | 1, 2 | eqtr3i 2135 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑} |
4 | difab 3309 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} | |
5 | abid2 2233 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ V} = V | |
6 | 5 | difeq1i 3154 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ V} ∖ {𝑥 ∣ 𝜑}) = (V ∖ {𝑥 ∣ 𝜑}) |
7 | 4, 6 | eqtr3i 2135 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥 ∣ 𝜑}) |
8 | 3, 7 | eqtr3i 2135 | 1 ⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1312 ∈ wcel 1461 {cab 2099 {crab 2392 Vcvv 2655 ∖ cdif 3032 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-rab 2397 df-v 2657 df-dif 3037 |
This theorem is referenced by: dfif3 3451 |
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