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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 3741 | . 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 2160 ⊆ wss 3144 {csn 3607 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-sn 3613 |
This theorem is referenced by: difsnss 3753 sssnm 3769 tpssi 3774 snelpwi 4230 intid 4242 abnexg 4464 ordsucss 4521 xpsspw 4756 djussxp 4790 xpimasn 5095 fconst6g 5433 f1sng 5522 fvimacnvi 5650 fsn2 5710 fnressn 5722 fsnunf 5736 mapsn 6715 unsnfidcel 6948 en1eqsn 6976 exmidfodomrlemim 7229 axresscn 7888 nn0ssre 9209 1fv 10168 fxnn0nninf 10468 1exp 10579 hashdifsn 10830 hashdifpr 10831 fsum00 11501 hash2iun1dif1 11519 4sqlem19 12440 exmidunben 12476 lspsncl 13705 lspsnss 13717 lspsnid 13720 znlidl 13927 isneip 14098 neipsm 14106 opnneip 14111 |
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