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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 3571 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
2 | 1 | ibi 174 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 ⊆ wss 2999 {csn 3444 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-in 3005 df-ss 3012 df-sn 3450 |
This theorem is referenced by: difsnss 3581 sssnm 3596 tpssi 3601 snelpwi 4037 intid 4049 ordsucss 4319 xpsspw 4546 djussxp 4577 xpimasn 4874 fconst6g 5203 fvimacnvi 5407 fsn2 5465 fnressn 5477 fsnunf 5490 mapsn 6437 unsnfidcel 6621 en1eqsn 6647 exmidfodomrlemim 6817 axresscn 7387 nn0ssre 8667 1fv 9538 fxnn0nninf 9832 1exp 9972 hashdifsn 10215 hashdifpr 10216 fsum00 10843 hash2iun1dif1 10861 |
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