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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 2200 | . 2 | |
2 | sneq 3533 | . . 3 | |
3 | 2 | sseq1d 3121 | . 2 |
4 | vex 2684 | . . 3 | |
5 | 4 | snss 3644 | . 2 |
6 | 1, 3, 5 | vtoclbg 2742 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wcel 1480 wss 3066 csn 3522 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-sn 3528 |
This theorem is referenced by: snssi 3659 snssd 3660 prssg 3672 ordtri2orexmid 4433 ordtri2or2exmid 4481 relsng 4637 fvimacnvi 5527 fvimacnv 5528 strslfv 11992 isneip 12304 elnei 12310 iscnp4 12376 cnpnei 12377 |
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