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Mirrors > Home > ILE Home > Th. List > nfdif | GIF version |
Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
nfdif.1 | ⊢ Ⅎ𝑥𝐴 |
nfdif.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfdif | ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdif2 3008 | . 2 ⊢ (𝐴 ∖ 𝐵) = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵} | |
2 | nfdif.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
3 | 2 | nfcri 2223 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
4 | 3 | nfn 1594 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵 |
5 | nfdif.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfrabxy 2548 | . 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2226 | 1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1439 Ⅎwnfc 2216 {crab 2364 ∖ cdif 2997 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rab 2369 df-dif 3002 |
This theorem is referenced by: (None) |
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