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Mirrors > Home > ILE Home > Th. List > sssnr | Unicode version |
Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4178. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
sssnr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3445 | . . 3 | |
2 | sseq1 3163 | . . 3 | |
3 | 1, 2 | mpbiri 167 | . 2 |
4 | eqimss 3194 | . 2 | |
5 | 3, 4 | jaoi 706 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 698 wceq 1342 wss 3114 c0 3407 csn 3573 |
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 2726 df-dif 3116 df-in 3120 df-ss 3127 df-nul 3408 |
This theorem is referenced by: pwsnss 3780 |
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