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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 5627 | . 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V)) | |
2 | snssg 4731 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))) | |
3 | 1, 2 | bitr4id 289 | 1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2105 Vcvv 3441 ⊆ wss 3898 {csn 4573 × cxp 5618 Rel wrel 5625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-in 3905 df-ss 3915 df-sn 4574 df-rel 5627 |
This theorem is referenced by: relsnb 5744 relsnopg 5745 relsn 5746 relsn2 6150 |
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