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Mirrors > Home > MPE Home > Th. List > relsnb | Structured version Visualization version GIF version |
Description: An at-most-singleton is a relation iff it is empty (because it is a "singleton on a proper class") or it is a singleton of an ordered pair. (Contributed by BJ, 26-Feb-2023.) |
Ref | Expression |
---|---|
relsnb | ⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsng 5814 | . . . 4 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
2 | 1 | biimpcd 249 | . . 3 ⊢ (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V))) |
3 | imor 853 | . . 3 ⊢ ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) | |
4 | 2, 3 | sylib 218 | . 2 ⊢ (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) |
5 | snprc 4722 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
6 | rel0 5812 | . . . . 5 ⊢ Rel ∅ | |
7 | releq 5789 | . . . . 5 ⊢ ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅)) | |
8 | 6, 7 | mpbiri 258 | . . . 4 ⊢ ({𝐴} = ∅ → Rel {𝐴}) |
9 | 5, 8 | sylbi 217 | . . 3 ⊢ (¬ 𝐴 ∈ V → Rel {𝐴}) |
10 | relsng 5814 | . . . 4 ⊢ (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
11 | 10 | ibir 268 | . . 3 ⊢ (𝐴 ∈ (V × V) → Rel {𝐴}) |
12 | 9, 11 | jaoi 857 | . 2 ⊢ ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴}) |
13 | 4, 12 | impbii 209 | 1 ⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 × cxp 5687 Rel wrel 5694 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-ss 3980 df-nul 4340 df-sn 4632 df-rel 5696 |
This theorem is referenced by: (None) |
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