![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4549 | . 2 ⊢ ¬ (𝐴 ∈ 𝒫 𝒫 ∪ 𝐴 ∧ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴) | |
2 | pwuni 4189 | . . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
3 | elpwg 3582 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 ∪ 𝐴 ↔ 𝐴 ⊆ 𝒫 ∪ 𝐴)) | |
4 | 2, 3 | mpbiri 168 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝐴) |
5 | ax-ia3 108 | . . 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 664 | 1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∈ wcel 2148 ⊆ wss 3129 𝒫 cpw 3574 ∪ cuni 3807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-setind 4532 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-uni 3808 |
This theorem is referenced by: mnfnre 7977 |
Copyright terms: Public domain | W3C validator |