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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwdifcl | Structured version Visualization version GIF version |
Description: Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
Ref | Expression |
---|---|
elpwincl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶) |
Ref | Expression |
---|---|
elpwdifcl | ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwincl.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶) | |
2 | 1 | elpwid 4614 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
3 | 2 | ssdifssd 4157 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶) |
4 | difexg 5335 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∖ 𝐵) ∈ V) | |
5 | elpwg 4608 | . . 3 ⊢ ((𝐴 ∖ 𝐵) ∈ V → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶)) | |
6 | 1, 4, 5 | 3syl 18 | . 2 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶)) |
7 | 3, 6 | mpbird 257 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-pw 4607 |
This theorem is referenced by: pwldsys 34138 ldgenpisyslem1 34144 difelcarsg 34292 inelcarsg 34293 carsgclctunlem2 34301 carsgclctunlem3 34302 carsgclctun 34303 |
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