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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 2926 | . 2 ⊢ (𝐴 ∖ 𝐵) = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵} | |
2 | nfdif.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
3 | 2 | nfcri 2172 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
4 | 3 | nfn 1548 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵 |
5 | nfdif.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfrabxy 2490 | . 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2175 | 1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1393 Ⅎwnfc 2165 {crab 2310 ∖ cdif 2914 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-dif 2920 |
This theorem is referenced by: (None) |
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