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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 4548 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
3 | 2 | ssdifssd 4083 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶) |
4 | difexg 5260 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∖ 𝐵) ∈ V) | |
5 | elpwg 4542 | . . 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 205 ∈ wcel 2104 Vcvv 3437 ∖ cdif 3889 ⊆ wss 3892 𝒫 cpw 4539 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 ax-sep 5232 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3287 df-v 3439 df-dif 3895 df-in 3899 df-ss 3909 df-pw 4541 |
This theorem is referenced by: pwldsys 32170 ldgenpisyslem1 32176 difelcarsg 32322 inelcarsg 32323 carsgclctunlem2 32331 carsgclctunlem3 32332 carsgclctun 32333 |
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