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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 3688 | . 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 1479 ∈ wcel 2135 Vcvv 2721 {csn 3570 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-sn 3576 |
This theorem is referenced by: mss 4198 ssfilem 6832 diffitest 6844 djuexb 7000 exmidonfinlem 7140 exmidfodomrlemim 7148 cc2lem 7198 |
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