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Mirrors > Home > ILE Home > Th. List > snm | GIF version |
Description: The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.) |
Ref | Expression |
---|---|
snnz.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snm | ⊢ ∃𝑥 𝑥 ∈ {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snnz.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | snmg 3712 | . 2 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 ∈ {𝐴}) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∃𝑥 𝑥 ∈ {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∃wex 1492 ∈ wcel 2148 Vcvv 2739 {csn 3594 |
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 2741 df-sn 3600 |
This theorem is referenced by: mss 4228 ssfilem 6878 diffitest 6890 djuexb 7046 exmidonfinlem 7195 exmidfodomrlemim 7203 cc2lem 7268 |
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