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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 3726 | . . 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 2148 ⊆ wss 3129 {csn 3592 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-sn 3598 |
This theorem is referenced by: pwntru 4199 ecinxp 6609 xpdom3m 6833 ac6sfi 6897 undifdc 6922 iunfidisj 6944 fidcenumlemr 6953 ssfii 6972 en2other2 7194 un0addcl 9208 un0mulcl 9209 fseq1p1m1 10093 fsumge1 11468 fprodsplit1f 11641 phicl2 12213 ennnfonelemhf1o 12413 imasaddfnlemg 12734 imasaddflemg 12736 0subm 12870 trivsubgd 13058 trivsubgsnd 13059 trivnsgd 13075 rest0 13615 iscnp4 13654 cnconst2 13669 cnpdis 13678 txdis 13713 txdis1cn 13714 fsumcncntop 13992 dvef 14124 bj-omtrans 14644 pwtrufal 14683 |
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