Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3641 | . 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 1468 ∈ wcel 1480 Vcvv 2686 {csn 3527 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sn 3533 |
This theorem is referenced by: mss 4148 ssfilem 6769 diffitest 6781 djuexb 6929 exmidonfinlem 7049 exmidfodomrlemim 7057 |
Copyright terms: Public domain | W3C validator |