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Mirrors > Home > MPE Home > Th. List > pwss | Structured version Visualization version GIF version |
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 | dfss2 3964 | . 2 ⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | velpw 4603 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
3 | 2 | imbi1i 349 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) |
4 | 3 | albii 1814 | . 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 205 ∀wal 1532 ∈ wcel 2099 ⊆ wss 3944 𝒫 cpw 4598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3471 df-in 3951 df-ss 3961 df-pw 4600 |
This theorem is referenced by: axpweq 5344 setind2 9750 axgroth5 10839 axgroth6 10843 grumnudlem 43645 ismnuprim 43654 |
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