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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 3626 | . 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 1465 ⊆ wss 3041 {csn 3497 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 df-sn 3503 |
This theorem is referenced by: difsnss 3636 sssnm 3651 tpssi 3656 snelpwi 4104 intid 4116 abnexg 4337 ordsucss 4390 xpsspw 4621 djussxp 4654 xpimasn 4957 fconst6g 5291 f1sng 5377 fvimacnvi 5502 fsn2 5562 fnressn 5574 fsnunf 5588 mapsn 6552 unsnfidcel 6777 en1eqsn 6804 exmidfodomrlemim 7025 axresscn 7636 nn0ssre 8949 1fv 9884 fxnn0nninf 10179 1exp 10290 hashdifsn 10533 hashdifpr 10534 fsum00 11199 hash2iun1dif1 11217 exmidunben 11866 isneip 12242 neipsm 12250 opnneip 12255 |
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