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Mirrors > Home > MPE Home > Th. List > pwnex | Structured version Visualization version GIF version |
Description: The class of all power sets is a proper class. See also snnex 7777. (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
pwnex | ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abnex 7776 | . . 3 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V) | |
2 | df-nel 3045 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V) | |
3 | 1, 2 | sylibr 234 | . 2 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V) |
4 | vpwex 5383 | . . 3 ⊢ 𝒫 𝑦 ∈ V | |
5 | vex 3482 | . . . 4 ⊢ 𝑦 ∈ V | |
6 | 5 | pwid 4627 | . . 3 ⊢ 𝑦 ∈ 𝒫 𝑦 |
7 | 4, 6 | pm3.2i 470 | . 2 ⊢ (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) |
8 | 3, 7 | mpg 1794 | 1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ∉ wnel 3044 Vcvv 3478 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-pow 5371 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-pw 4607 df-sn 4632 df-uni 4913 df-iun 4998 |
This theorem is referenced by: topnex 23019 |
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