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Mirrors > Home > ILE Home > Th. List > snssg | Unicode version |
Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) |
Ref | Expression |
---|---|
snssg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2233 | . 2 | |
2 | sneq 3592 | . . 3 | |
3 | 2 | sseq1d 3176 | . 2 |
4 | vex 2733 | . . 3 | |
5 | 4 | snss 3707 | . 2 |
6 | 1, 3, 5 | vtoclbg 2791 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1348 wcel 2141 wss 3121 csn 3581 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-sn 3587 |
This theorem is referenced by: snssi 3722 snssd 3723 prssg 3735 ordtri2orexmid 4505 ordtri2or2exmid 4553 ontri2orexmidim 4554 relsng 4712 fvimacnvi 5607 fvimacnv 5608 strslfv 12447 isneip 12899 elnei 12905 iscnp4 12971 cnpnei 12972 |
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