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Mirrors > Home > ILE Home > Th. List > snssi | GIF version |
Description: The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.) |
Ref | Expression |
---|---|
snssi | ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssg 3664 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
2 | 1 | ibi 175 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ⊆ wss 3076 {csn 3532 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 df-sn 3538 |
This theorem is referenced by: difsnss 3674 sssnm 3689 tpssi 3694 snelpwi 4142 intid 4154 abnexg 4375 ordsucss 4428 xpsspw 4659 djussxp 4692 xpimasn 4995 fconst6g 5329 f1sng 5417 fvimacnvi 5542 fsn2 5602 fnressn 5614 fsnunf 5628 mapsn 6592 unsnfidcel 6817 en1eqsn 6844 exmidfodomrlemim 7074 axresscn 7692 nn0ssre 9005 1fv 9947 fxnn0nninf 10242 1exp 10353 hashdifsn 10597 hashdifpr 10598 fsum00 11263 hash2iun1dif1 11281 exmidunben 11975 isneip 12354 neipsm 12362 opnneip 12367 |
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