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Mirrors > Home > MPE Home > Th. List > pwpwab | Structured version Visualization version GIF version |
Description: The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.) |
Ref | Expression |
---|---|
pwpwab | ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3497 | . . 3 ⊢ 𝑥 ∈ V | |
2 | elpwpw 5024 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴)) | |
3 | 1, 2 | mpbiran 707 | . 2 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴) |
4 | 3 | abbi2i 2953 | 1 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {cab 2799 Vcvv 3494 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-v 3496 df-in 3943 df-ss 3952 df-pw 4541 df-uni 4839 |
This theorem is referenced by: pwpwssunieq 5026 |
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