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Mirrors > Home > MPE Home > Th. List > eldifpw | Structured version Visualization version GIF version |
Description: Membership in a power class difference. (Contributed by NM, 25-Mar-2007.) |
Ref | Expression |
---|---|
eldifpw.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
eldifpw | ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4547 | . . . 4 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | unss1 4152 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | |
3 | eldifpw.1 | . . . . . . 7 ⊢ 𝐶 ∈ V | |
4 | unexg 7461 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ∪ 𝐶) ∈ V) | |
5 | 3, 4 | mpan2 687 | . . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ∈ V) |
6 | elpwg 4541 | . . . . . 6 ⊢ ((𝐴 ∪ 𝐶) ∈ V → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝒫 𝐵 → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))) |
8 | 2, 7 | syl5ibr 247 | . . . 4 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶))) |
9 | 1, 8 | mpd 15 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶)) |
10 | elpwi 4547 | . . . . 5 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ⊆ 𝐵) | |
11 | 10 | unssbd 4161 | . . . 4 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → 𝐶 ⊆ 𝐵) |
12 | 11 | con3i 157 | . . 3 ⊢ (¬ 𝐶 ⊆ 𝐵 → ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵) |
13 | 9, 12 | anim12i 612 | . 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵)) |
14 | eldif 3943 | . 2 ⊢ ((𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵) ↔ ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵)) | |
15 | 13, 14 | sylibr 235 | 1 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ∪ cun 3931 ⊆ wss 3933 𝒫 cpw 4535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-pw 4537 df-sn 4558 df-pr 4560 df-uni 4831 |
This theorem is referenced by: (None) |
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