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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 7670. (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
pwnex | ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abnex 7669 | . . 3 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V) | |
2 | df-nel 3047 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V) | |
3 | 1, 2 | sylibr 233 | . 2 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V) |
4 | vpwex 5320 | . . 3 ⊢ 𝒫 𝑦 ∈ V | |
5 | vex 3445 | . . . 4 ⊢ 𝑦 ∈ V | |
6 | 5 | pwid 4569 | . . 3 ⊢ 𝑦 ∈ 𝒫 𝑦 |
7 | 4, 6 | pm3.2i 471 | . 2 ⊢ (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) |
8 | 3, 7 | mpg 1798 | 1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∀wal 1538 = wceq 1540 ∃wex 1780 ∈ wcel 2105 {cab 2713 ∉ wnel 3046 Vcvv 3441 𝒫 cpw 4547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-11 2153 ax-ext 2707 ax-sep 5243 ax-pow 5308 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-in 3905 df-ss 3915 df-pw 4549 df-sn 4574 df-uni 4853 df-iun 4943 |
This theorem is referenced by: topnex 22252 |
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