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Mirrors > Home > MPE Home > Th. List > pwpwssunieq | Structured version Visualization version GIF version |
Description: The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.) |
Ref | Expression |
---|---|
pwpwssunieq | ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 4032 | . . 3 ⊢ (∪ 𝑥 = 𝐴 → ∪ 𝑥 ⊆ 𝐴) | |
2 | 1 | ss2abi 4055 | . 2 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} |
3 | pwpwab 5096 | . 2 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} | |
4 | 2, 3 | sseqtrri 4011 | 1 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 {cab 2701 ⊆ wss 3940 𝒫 cpw 4594 ∪ cuni 4899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-v 3468 df-in 3947 df-ss 3957 df-pw 4596 df-uni 4900 |
This theorem is referenced by: toponsspwpw 22746 dmtopon 22747 |
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