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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 | snssg 4716 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))) | |
2 | df-rel 5561 | . 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V)) | |
3 | 1, 2 | syl6rbbr 292 | 1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 {csn 4566 × cxp 5552 Rel wrel 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3942 df-ss 3951 df-sn 4567 df-rel 5561 |
This theorem is referenced by: relsnb 5674 relsnopg 5675 relsn 5676 relsn2 6068 |
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