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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 3716 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) |
4 | 1, 3 | mpbid 146 | 1 ⊢ (𝜑 → {𝐴} ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 2141 ⊆ wss 3121 {csn 3583 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-sn 3589 |
This theorem is referenced by: pwntru 4185 ecinxp 6588 xpdom3m 6812 ac6sfi 6876 undifdc 6901 iunfidisj 6923 fidcenumlemr 6932 ssfii 6951 en2other2 7173 un0addcl 9168 un0mulcl 9169 fseq1p1m1 10050 fsumge1 11424 fprodsplit1f 11597 phicl2 12168 ennnfonelemhf1o 12368 0subm 12702 rest0 12973 iscnp4 13012 cnconst2 13027 cnpdis 13036 txdis 13071 txdis1cn 13072 fsumcncntop 13350 dvef 13482 bj-omtrans 13991 pwtrufal 14030 |
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