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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 3682 |
. . 3
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2 | simpl 108 |
. . . 4
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3 | eleq1 2217 |
. . . . . . . 8
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4 | 3 | biimpcd 158 |
. . . . . . 7
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5 | 4 | necon3bd 2367 |
. . . . . 6
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6 | 5 | com12 30 |
. . . . 5
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7 | 6 | ancld 323 |
. . . 4
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8 | 2, 7 | impbid2 142 |
. . 3
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9 | 1, 8 | syl5bb 191 |
. 2
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10 | 9 | eqrdv 2152 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1740 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-v 2711 df-dif 3100 df-sn 3562 |
This theorem is referenced by: difsnb 3695 fisseneq 6865 dfn2 9082 |
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