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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 3459 | . 2 ⊢ (𝐴 ∈ 𝒫 (V × V) → 𝐴 ∈ V) | |
2 | pwvrel 5566 | . 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 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 𝒫 cpw 4497 × cxp 5517 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-pw 4499 df-rel 5526 |
This theorem is referenced by: (None) |
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