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Mirrors > Home > MPE Home > Th. List > difelpw | Structured version Visualization version GIF version |
Description: A difference is an element of the power set of its minuend. (Contributed by AV, 9-Oct-2023.) |
Ref | Expression |
---|---|
difelpw | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4108 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
2 | elpw2g 5247 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐴)) | |
3 | 1, 2 | mpbiri 260 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∖ cdif 3933 ⊆ wss 3936 𝒫 cpw 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-in 3943 df-ss 3952 df-pw 4541 |
This theorem is referenced by: satfvsuclem2 32607 clsk3nimkb 40439 |
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