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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 3611 | . 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 1453 ∈ wcel 1465 Vcvv 2660 {csn 3497 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-sn 3503 |
This theorem is referenced by: mss 4118 ssfilem 6737 diffitest 6749 djuexb 6897 exmidonfinlem 7017 exmidfodomrlemim 7025 |
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