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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 4565 | . 2 ⊢ ¬ (𝐴 ∈ 𝒫 𝒫 ∪ 𝐴 ∧ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴) | |
2 | pwuni 4204 | . . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
3 | elpwg 3595 | . . . 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 2158 ⊆ wss 3141 𝒫 cpw 3587 ∪ cuni 3821 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-setind 4548 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-uni 3822 |
This theorem is referenced by: mnfnre 8013 |
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