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| Description: Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) | 
| Ref | Expression | 
|---|---|
| pwss | ⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ss 3967 | . 2 ⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵)) | |
| 2 | velpw 4604 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
| 3 | 2 | imbi1i 349 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | 
| 4 | 3 | albii 1818 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | 
| 5 | 1, 4 | bitri 275 | 1 ⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-pw 4601 | 
| This theorem is referenced by: axpweq 5350 setind2 9776 axgroth5 10865 axgroth6 10869 grumnudlem 44309 ismnuprim 44318 | 
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