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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 4386 | . 2 ⊢ (Rel {A} ↔ A ∈ (V × V)) |
3 | dmsnm 4729 | . 2 ⊢ (A ∈ (V × V) ↔ ∃x x ∈ dom {A}) | |
4 | 2, 3 | bitri 173 | 1 ⊢ (Rel {A} ↔ ∃x x ∈ dom {A}) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 {csn 3367 × cxp 4286 dom cdm 4288 Rel wrel 4293 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-dm 4298 |
This theorem is referenced by: (None) |
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