![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > relsn2m | GIF version |
Description: A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Ref | Expression |
---|---|
relsn2m.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
relsn2m | ⊢ (Rel {𝐴} ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsn2m.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | relsn 4539 | . 2 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V)) |
3 | dmsnm 4891 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
4 | 2, 3 | bitri 182 | 1 ⊢ (Rel {𝐴} ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∃wex 1426 ∈ wcel 1438 Vcvv 2619 {csn 3444 × cxp 4434 dom cdm 4436 Rel wrel 4441 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-br 3844 df-opab 3898 df-xp 4442 df-rel 4443 df-dm 4446 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |