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Mirrors > Home > MPE Home > Th. List > relsn2 | Structured version Visualization version GIF version |
Description: A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.) Make hypothesis an antecedent. (Revised by BJ, 12-Feb-2022.) |
Ref | Expression |
---|---|
relsn2 | ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsng 5669 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
2 | dmsnn0 6059 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
3 | 1, 2 | syl6bb 289 | 1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 ∅c0 4291 {csn 4561 × cxp 5548 dom cdm 5550 Rel wrel 5555 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
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-ne 3017 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-dm 5560 |
This theorem is referenced by: (None) |
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