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Mirrors > Home > MPE Home > Th. List > elpwun | Structured version Visualization version GIF version |
Description: Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.) |
Ref | Expression |
---|---|
eldifpw.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elpwun | ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3490 | . 2 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) → 𝐴 ∈ V) | |
2 | elex 3490 | . . 3 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∖ 𝐶) ∈ V) | |
3 | eldifpw.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | difex2 7762 | . . . 4 ⊢ (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V) |
6 | 2, 5 | sylibr 233 | . 2 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → 𝐴 ∈ V) |
7 | elpwg 4606 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∪ 𝐶))) | |
8 | uncom 4152 | . . . . . 6 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
9 | 8 | sseq2i 4009 | . . . . 5 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
10 | ssundif 4488 | . . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∪ 𝐵) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) | |
11 | 9, 10 | bitri 275 | . . . 4 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) |
12 | difexg 5329 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐶) ∈ V) | |
13 | elpwg 4606 | . . . . 5 ⊢ ((𝐴 ∖ 𝐶) ∈ V → ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝐴 ∈ V → ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)) |
15 | 11, 14 | bitr4id 290 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)) |
16 | 7, 15 | bitrd 279 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)) |
17 | 1, 6, 16 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2099 Vcvv 3471 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 𝒫 cpw 4603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-pw 4605 df-sn 4630 df-pr 4632 df-uni 4909 |
This theorem is referenced by: pwfilem 9202 pwfilemOLD 9371 elrfi 42114 dssmapnvod 43450 |
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