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| Mirrors > Home > ILE Home > Th. List > snssd | GIF version | ||
| Description: The singleton of an element of a class is a subset of the class (deduction form). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
| Ref | Expression |
|---|---|
| snssd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| snssd | ⊢ (𝜑 → {𝐴} ⊆ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snssd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
| 2 | snssg 3766 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
| 3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) |
| 4 | 1, 3 | mpbid 147 | 1 ⊢ (𝜑 → {𝐴} ⊆ 𝐵) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2175 ⊆ wss 3165 {csn 3632 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-v 2773 df-in 3171 df-ss 3178 df-sn 3638 |
| This theorem is referenced by: pwntru 4242 ecinxp 6696 xpdom3m 6928 ac6sfi 6994 undifdc 7020 iunfidisj 7047 fidcenumlemr 7056 ssfii 7075 en2other2 7303 un0addcl 9327 un0mulcl 9328 fseq1p1m1 10215 fsumge1 11743 fprodsplit1f 11916 bitsinv1 12244 phicl2 12507 ennnfonelemhf1o 12755 imasaddfnlemg 13117 imasaddflemg 13119 0subm 13287 gsumvallem2 13296 trivsubgd 13507 trivsubgsnd 13508 trivnsgd 13524 kerf1ghm 13581 lsssn0 14103 lss0ss 14104 lsptpcl 14127 lspsnvsi 14151 lspun0 14158 mulgrhm2 14343 zndvds 14382 rest0 14622 iscnp4 14661 cnconst2 14676 cnpdis 14685 txdis 14720 txdis1cn 14721 fsumcncntop 15010 dvef 15170 plyf 15180 elplyr 15183 elplyd 15184 ply1term 15186 plyaddlem 15192 plymullem 15193 plycolemc 15201 plycn 15205 dvply2g 15209 perfectlem2 15443 bj-omtrans 15854 pwtrufal 15896 |
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