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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 5463 | . . . 4 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
2 | 1 | biimpcd 241 | . . 3 ⊢ (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V))) |
3 | imor 884 | . . 3 ⊢ ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) | |
4 | 2, 3 | sylib 210 | . 2 ⊢ (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) |
5 | snprc 4473 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
6 | rel0 5461 | . . . . 5 ⊢ Rel ∅ | |
7 | releq 5440 | . . . . 5 ⊢ ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅)) | |
8 | 6, 7 | mpbiri 250 | . . . 4 ⊢ ({𝐴} = ∅ → Rel {𝐴}) |
9 | 5, 8 | sylbi 209 | . . 3 ⊢ (¬ 𝐴 ∈ V → Rel {𝐴}) |
10 | relsng 5463 | . . . 4 ⊢ (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
11 | 10 | ibir 260 | . . 3 ⊢ (𝐴 ∈ (V × V) → Rel {𝐴}) |
12 | 9, 11 | jaoi 888 | . 2 ⊢ ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴}) |
13 | 4, 12 | impbii 201 | 1 ⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∨ wo 878 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∅c0 4146 {csn 4399 × cxp 5344 Rel wrel 5351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 df-dif 3801 df-in 3805 df-ss 3812 df-nul 4147 df-sn 4400 df-rel 5353 |
This theorem is referenced by: (None) |
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