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Mirrors > Home > ILE Home > Th. List > 2pwuninelg | GIF version |
Description: The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.) |
Ref | Expression |
---|---|
2pwuninelg | ⊢ (A ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ A ∈ A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2lp 4232 | . 2 ⊢ ¬ (A ∈ 𝒫 𝒫 ∪ A ∧ 𝒫 𝒫 ∪ A ∈ A) | |
2 | pwuni 3934 | . . . 4 ⊢ A ⊆ 𝒫 ∪ A | |
3 | elpwg 3359 | . . . 4 ⊢ (A ∈ 𝑉 → (A ∈ 𝒫 𝒫 ∪ A ↔ A ⊆ 𝒫 ∪ A)) | |
4 | 2, 3 | mpbiri 157 | . . 3 ⊢ (A ∈ 𝑉 → A ∈ 𝒫 𝒫 ∪ A) |
5 | ax-ia3 101 | . . 3 ⊢ (A ∈ 𝒫 𝒫 ∪ A → (𝒫 𝒫 ∪ A ∈ A → (A ∈ 𝒫 𝒫 ∪ A ∧ 𝒫 𝒫 ∪ A ∈ A))) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (A ∈ 𝑉 → (𝒫 𝒫 ∪ A ∈ A → (A ∈ 𝒫 𝒫 ∪ A ∧ 𝒫 𝒫 ∪ A ∈ A))) |
7 | 1, 6 | mtoi 589 | 1 ⊢ (A ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ A ∈ A) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∈ wcel 1390 ⊆ wss 2911 𝒫 cpw 3351 ∪ cuni 3571 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-uni 3572 |
This theorem is referenced by: mnfnre 6865 |
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