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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 4586 | . 2 ⊢ ¬ (𝐴 ∈ 𝒫 𝒫 ∪ 𝐴 ∧ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴) | |
2 | pwuni 4221 | . . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
3 | elpwg 3609 | . . . 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 665 | 1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∈ wcel 2164 ⊆ wss 3153 𝒫 cpw 3601 ∪ cuni 3835 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 ax-setind 4569 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-uni 3836 |
This theorem is referenced by: mnfnre 8062 |
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