Theorem pwss 4572

Description: Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)

Assertion

Ref	Expression
pwss	⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pwss

Step	Hyp	Ref	Expression
1		df-ss 3915	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵))
2		velpw 4554	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	imbi1i 349	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵))
4	3	albii 1820	. 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))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   ∈ wcel 2113   ⊆ wss 3898  𝒫 cpw 4549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-ss 3915  df-pw 4551

This theorem is referenced by:  axpweq  5291  setind2  9645  axgroth5  10722  axgroth6  10726  grumnudlem  44402  ismnuprim  44411