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Mirrors > Home > ILE Home > Th. List > pwnex | GIF version |
Description: The class of all power sets is a proper class. See also snnex 4469. (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
pwnex | ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abnex 4468 | . . 3 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V) | |
2 | df-nel 2456 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V) | |
3 | 1, 2 | sylibr 134 | . 2 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V) |
4 | vpwex 4200 | . . 3 ⊢ 𝒫 𝑦 ∈ V | |
5 | vex 2755 | . . . 4 ⊢ 𝑦 ∈ V | |
6 | 5 | pwid 3608 | . . 3 ⊢ 𝑦 ∈ 𝒫 𝑦 |
7 | 4, 6 | pm3.2i 272 | . 2 ⊢ (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) |
8 | 3, 7 | mpg 1462 | 1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ∀wal 1362 = wceq 1364 ∃wex 1503 ∈ wcel 2160 {cab 2175 ∉ wnel 2455 Vcvv 2752 𝒫 cpw 3593 |
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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-un 4454 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-nel 2456 df-ral 2473 df-rex 2474 df-v 2754 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-uni 3828 df-iun 3906 |
This theorem is referenced by: topnex 14071 |
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