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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 3500 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
2 | 1 | ibi 165 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ⊆ wss 2917 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-sn 3381 |
This theorem is referenced by: difsnss 3510 sssnm 3525 tpssi 3530 snelpwi 3948 intid 3960 ordsucss 4230 xpsspw 4450 djussxp 4481 xpimasn 4769 fconst6g 5085 fvimacnvi 5281 fsn2 5337 fnressn 5349 fsnunf 5362 axresscn 6936 nn0ssre 8185 1fv 8996 1exp 9284 |
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