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Mirrors > Home > ILE Home > Th. List > difsnss | Unicode version |
Description: If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6466. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
difsnss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3261 | . 2 | |
2 | snssi 3711 | . . 3 | |
3 | undifss 3484 | . . 3 | |
4 | 2, 3 | sylib 121 | . 2 |
5 | 1, 4 | eqsstrid 3183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2135 cdif 3108 cun 3109 wss 3111 csn 3570 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 |
This theorem is referenced by: fnsnsplitss 5678 dcdifsnid 6463 |
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