Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > relsng | Structured version Visualization version GIF version |
Description: A singleton is a relation iff it is a singleton on an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.) |
Ref | Expression |
---|---|
relsng | ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5573 | . 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V)) | |
2 | snssg 4713 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))) | |
3 | 1, 2 | bitr4id 293 | 1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 Vcvv 3421 ⊆ wss 3881 {csn 4556 × cxp 5564 Rel wrel 5571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-v 3423 df-in 3888 df-ss 3898 df-sn 4557 df-rel 5573 |
This theorem is referenced by: relsnb 5687 relsnopg 5688 relsn 5689 relsn2 6090 |
Copyright terms: Public domain | W3C validator |