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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 3509 | . 2 ⊢ (𝐴 ∈ 𝒫 (V × V) → 𝐴 ∈ V) | |
2 | pwvrel 5595 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴)) | |
3 | 1, 2 | biadanii 820 | 1 ⊢ (𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2113 Vcvv 3491 𝒫 cpw 4532 × 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-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-in 3936 df-ss 3945 df-pw 4534 df-rel 5555 |
This theorem is referenced by: (None) |
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