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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 3577 | . . . 4 | |
2 | 1 | imbi1i 237 | . . 3 |
3 | 2 | albii 1450 | . 2 |
4 | dfss2 3117 | . 2 | |
5 | snss.1 | . . 3 | |
6 | 5 | clel2 2845 | . 2 |
7 | 3, 4, 6 | 3bitr4ri 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wal 1333 wceq 1335 wcel 2128 cvv 2712 wss 3102 csn 3560 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-in 3108 df-ss 3115 df-sn 3566 |
This theorem is referenced by: snssg 3692 prss 3712 tpss 3721 snelpw 4173 sspwb 4176 mss 4186 exss 4187 reg2exmidlema 4492 elomssom 4563 relsn 4690 fnressn 5652 un0mulcl 9119 nn0ssz 9180 fimaxre2 11121 fsum2dlemstep 11326 fsumabs 11357 fsumiun 11369 fprod2dlemstep 11514 bdsnss 13435 |
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