Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7480. (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
pwnex | ⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abnex 7479 | . . 3 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V) | |
2 | df-nel 3124 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V) | |
3 | 1, 2 | sylibr 236 | . 2 ⊢ (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V) |
4 | vpwex 5278 | . . 3 ⊢ 𝒫 𝑦 ∈ V | |
5 | vex 3497 | . . . 4 ⊢ 𝑦 ∈ V | |
6 | 5 | pwid 4563 | . . 3 ⊢ 𝑦 ∈ 𝒫 𝑦 |
7 | 4, 6 | pm3.2i 473 | . 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 398 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ∉ wnel 3123 Vcvv 3494 𝒫 cpw 4539 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-pow 5266 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-pw 4541 df-sn 4568 df-uni 4839 df-iun 4921 |
This theorem is referenced by: topnex 21604 |
Copyright terms: Public domain | W3C validator |