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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 5811 | . . . 4 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
| 2 | 1 | biimpcd 249 | . . 3 ⊢ (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V))) | 
| 3 | imor 854 | . . 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 4717 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 6 | rel0 5809 | . . . . 5 ⊢ Rel ∅ | |
| 7 | releq 5786 | . . . . 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 5811 | . . . 4 ⊢ (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
| 11 | 10 | ibir 268 | . . 3 ⊢ (𝐴 ∈ (V × V) → Rel {𝐴}) | 
| 12 | 9, 11 | jaoi 858 | . 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 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 × cxp 5683 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-sn 4627 df-rel 5692 | 
| This theorem is referenced by: (None) | 
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