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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 5100 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | |
| 2 | unissb 4939 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | |
| 3 | 1, 2 | bitri 275 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wral 3061 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-ss 3968 df-pw 4602 df-uni 4908 |
| This theorem is referenced by: ustuni 24235 metustfbas 24570 intlidl 33448 dmvlsiga 34130 1stmbfm 34262 2ndmbfm 34263 dya2iocucvr 34286 gneispace 44147 preimafvsspwdm 47376 usgrexmpl1lem 47980 usgrexmpl2lem 47985 |
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