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Mirrors > Home > ILE Home > Th. List > snss | 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, 5-Aug-1993.) |
Ref | Expression |
---|---|
snss.1 |
Ref | Expression |
---|---|
snss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | velsn 3600 | . . . 4 | |
2 | 1 | imbi1i 237 | . . 3 |
3 | 2 | albii 1463 | . 2 |
4 | dfss2 3136 | . 2 | |
5 | snss.1 | . . 3 | |
6 | 5 | clel2 2863 | . 2 |
7 | 3, 4, 6 | 3bitr4ri 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wal 1346 wceq 1348 wcel 2141 cvv 2730 wss 3121 csn 3583 |
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 3589 |
This theorem is referenced by: snssg 3716 prss 3736 tpss 3745 snelpw 4198 sspwb 4201 mss 4211 exss 4212 reg2exmidlema 4518 elomssom 4589 relsn 4716 fnressn 5682 un0mulcl 9169 nn0ssz 9230 fimaxre2 11190 fsum2dlemstep 11397 fsumabs 11428 fsumiun 11440 fprod2dlemstep 11585 bdsnss 13908 |
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