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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 3588 | . 2 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 ∈ {𝐴}) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ ∃𝑥 𝑥 ∈ {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∃wex 1436 ∈ wcel 1448 Vcvv 2641 {csn 3474 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-sn 3480 |
This theorem is referenced by: mss 4086 ssfilem 6698 diffitest 6710 djuexb 6844 exmidfodomrlemim 6966 |
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