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Mirrors > Home > NFE 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 3839 | . . 3 | |
2 | simpl 443 | . . . 4 | |
3 | eleq1 2413 | . . . . . . . 8 | |
4 | 3 | biimpcd 215 | . . . . . . 7 |
5 | 4 | necon3bd 2553 | . . . . . 6 |
6 | 5 | com12 27 | . . . . 5 |
7 | 6 | ancld 536 | . . . 4 |
8 | 2, 7 | impbid2 195 | . . 3 |
9 | 1, 8 | syl5bb 248 | . 2 |
10 | 9 | eqrdv 2351 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 358 wceq 1642 wcel 1710 wne 2516 cdif 3206 csn 3737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-sn 3741 |
This theorem is referenced by: difsnb 3850 adj11 3889 nnsucelrlem3 4426 nnsucelr 4428 ssfin 4470 |
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