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Mirrors > Home > MPE Home > Th. List > pwssb | Structured version Visualization version GIF version |
Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.) |
Ref | Expression |
---|---|
pwssb | ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwuni 4985 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | |
2 | unissb 4832 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | |
3 | 1, 2 | bitri 278 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wral 3106 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-11 2158 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-pw 4499 df-uni 4801 |
This theorem is referenced by: ustuni 22832 metustfbas 23164 intlidl 31010 dmvlsiga 31498 1stmbfm 31628 2ndmbfm 31629 dya2iocucvr 31652 gneispace 40837 preimafvsspwdm 43906 |
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