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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 4040 | . . 3 ⊢ (∪ 𝑥 = 𝐴 → ∪ 𝑥 ⊆ 𝐴) | |
2 | 1 | ss2abi 4063 | . 2 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} |
3 | pwpwab 5106 | . 2 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} | |
4 | 2, 3 | sseqtrri 4019 | 1 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {cab 2710 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-v 3477 df-in 3955 df-ss 3965 df-pw 4604 df-uni 4909 |
This theorem is referenced by: toponsspwpw 22416 dmtopon 22417 |
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