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Mirrors > Home > ILE Home > Th. List > difsn | Unicode version |
Description: An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
difsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 3495 | . . 3 | |
2 | simpl 102 | . . . 4 | |
3 | eleq1 2100 | . . . . . . . 8 | |
4 | 3 | biimpcd 148 | . . . . . . 7 |
5 | 4 | necon3bd 2248 | . . . . . 6 |
6 | 5 | com12 27 | . . . . 5 |
7 | 6 | ancld 308 | . . . 4 |
8 | 2, 7 | impbid2 131 | . . 3 |
9 | 1, 8 | syl5bb 181 | . 2 |
10 | 9 | eqrdv 2038 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wceq 1243 wcel 1393 wne 2204 cdif 2914 csn 3375 |
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-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-v 2559 df-dif 2920 df-sn 3381 |
This theorem is referenced by: difsnb 3506 dfn2 8194 |
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