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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 5105 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | |
2 | unissb 4944 | . 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 3059 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-ss 3980 df-pw 4607 df-uni 4913 |
This theorem is referenced by: ustuni 24251 metustfbas 24586 intlidl 33428 dmvlsiga 34110 1stmbfm 34242 2ndmbfm 34243 dya2iocucvr 34266 gneispace 44124 preimafvsspwdm 47314 usgrexmpl1lem 47916 usgrexmpl2lem 47921 |
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