![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > pwelg | Structured version Visualization version GIF version |
Description: The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.) |
Ref | Expression |
---|---|
pwelg | ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → 𝒫 𝑥 ∈ 𝐵) | |
2 | 1 | ralimi 3075 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 𝒫 𝑥 ∈ 𝐵) |
3 | pweq 4609 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
4 | 3 | eleq1d 2810 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵)) |
5 | 4 | rspccv 3601 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝒫 𝑥 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝐵)) |
6 | 2, 5 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝐵)) |
7 | simpl 482 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐵) | |
8 | 7 | ralimi 3075 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∪ 𝑥 ∈ 𝐵) |
9 | unieq 4911 | . . . . . 6 ⊢ (𝑥 = 𝒫 𝐴 → ∪ 𝑥 = ∪ 𝒫 𝐴) | |
10 | unipw 5441 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
11 | 9, 10 | eqtrdi 2780 | . . . . 5 ⊢ (𝑥 = 𝒫 𝐴 → ∪ 𝑥 = 𝐴) |
12 | 11 | eleq1d 2810 | . . . 4 ⊢ (𝑥 = 𝒫 𝐴 → (∪ 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
13 | 12 | rspccv 3601 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∪ 𝑥 ∈ 𝐵 → (𝒫 𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐵)) |
14 | 8, 13 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝒫 𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐵)) |
15 | 6, 14 | impbid 211 | 1 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 𝒫 cpw 4595 ∪ cuni 4900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-v 3468 df-un 3946 df-in 3948 df-ss 3958 df-pw 4597 df-sn 4622 df-pr 4624 df-uni 4901 |
This theorem is referenced by: pwinfig 42862 |
Copyright terms: Public domain | W3C validator |