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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 3416 | . 2 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) → 𝐴 ∈ V) | |
2 | elex 3416 | . . 3 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∖ 𝐶) ∈ V) | |
3 | eldifpw.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | difex2 7523 | . . . 4 ⊢ (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V) |
6 | 2, 5 | sylibr 237 | . 2 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → 𝐴 ∈ V) |
7 | elpwg 4502 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∪ 𝐶))) | |
8 | uncom 4053 | . . . . . 6 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
9 | 8 | sseq2i 3916 | . . . . 5 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
10 | ssundif 4385 | . . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∪ 𝐵) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) | |
11 | 9, 10 | bitri 278 | . . . 4 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) |
12 | difexg 5205 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐶) ∈ V) | |
13 | elpwg 4502 | . . . . 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 293 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)) |
16 | 7, 15 | bitrd 282 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)) |
17 | 1, 6, 16 | pm5.21nii 383 | 1 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2112 Vcvv 3398 ∖ cdif 3850 ∪ cun 3851 ⊆ wss 3853 𝒫 cpw 4499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-pw 4501 df-sn 4528 df-pr 4530 df-uni 4806 |
This theorem is referenced by: pwfilem 8832 pwfilemOLD 8948 elrfi 40160 dssmapnvod 41246 |
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