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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 4022 | . . 3 ⊢ (∪ 𝑥 = 𝐴 → ∪ 𝑥 ⊆ 𝐴) | |
2 | 1 | ss2abi 4042 | . 2 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} |
3 | pwpwab 5024 | . 2 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} | |
4 | 2, 3 | sseqtrri 4003 | 1 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 {cab 2799 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4837 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-in 3942 df-ss 3951 df-pw 4540 df-uni 4838 |
This theorem is referenced by: toponsspwpw 21529 dmtopon 21530 |
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