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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pwvrelb | Structured version Visualization version GIF version | ||
| Description: Characterization of the elements of the powerclass of the cartesian square of the universal class: they are exactly the sets which are binary relations. (Contributed by BJ, 16-Dec-2023.) | 
| Ref | Expression | 
|---|---|
| bj-pwvrelb | ⊢ (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elex 3500 | . 2 ⊢ (𝐴 ∈ 𝒫 (V × V) → 𝐴 ∈ V) | |
| 2 | pwvrel 5734 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴)) | |
| 3 | 1, 2 | biadanii 821 | 1 ⊢ (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2107 Vcvv 3479 𝒫 cpw 4599 × cxp 5682 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-pw 4601 df-rel 5691 | 
| This theorem is referenced by: (None) | 
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