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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 3725 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)) | |
2 | 1 | ibi 176 | 1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ⊆ wss 3129 {csn 3591 |
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 3597 |
This theorem is referenced by: difsnss 3737 sssnm 3752 tpssi 3757 snelpwi 4209 intid 4221 abnexg 4443 ordsucss 4500 xpsspw 4735 djussxp 4768 xpimasn 5073 fconst6g 5410 f1sng 5499 fvimacnvi 5626 fsn2 5686 fnressn 5698 fsnunf 5712 mapsn 6684 unsnfidcel 6914 en1eqsn 6941 exmidfodomrlemim 7194 axresscn 7847 nn0ssre 9166 1fv 10122 fxnn0nninf 10421 1exp 10532 hashdifsn 10780 hashdifpr 10781 fsum00 11451 hash2iun1dif1 11469 exmidunben 12407 isneip 13306 neipsm 13314 opnneip 13319 |
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