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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 5667 | . . . 4 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
2 | 1 | biimpcd 251 | . . 3 ⊢ (Rel {𝐴} → (𝐴 ∈ V → 𝐴 ∈ (V × V))) |
3 | imor 849 | . . 3 ⊢ ((𝐴 ∈ V → 𝐴 ∈ (V × V)) ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) | |
4 | 2, 3 | sylib 220 | . 2 ⊢ (Rel {𝐴} → (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) |
5 | snprc 4646 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
6 | rel0 5665 | . . . . 5 ⊢ Rel ∅ | |
7 | releq 5644 | . . . . 5 ⊢ ({𝐴} = ∅ → (Rel {𝐴} ↔ Rel ∅)) | |
8 | 6, 7 | mpbiri 260 | . . . 4 ⊢ ({𝐴} = ∅ → Rel {𝐴}) |
9 | 5, 8 | sylbi 219 | . . 3 ⊢ (¬ 𝐴 ∈ V → Rel {𝐴}) |
10 | relsng 5667 | . . . 4 ⊢ (𝐴 ∈ (V × V) → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
11 | 10 | ibir 270 | . . 3 ⊢ (𝐴 ∈ (V × V) → Rel {𝐴}) |
12 | 9, 11 | jaoi 853 | . 2 ⊢ ((¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)) → Rel {𝐴}) |
13 | 4, 12 | impbii 211 | 1 ⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 {csn 4560 × cxp 5546 Rel wrel 5553 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-dif 3932 df-in 3936 df-ss 3945 df-nul 4285 df-sn 4561 df-rel 5555 |
This theorem is referenced by: (None) |
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