Theorem pwss 4578

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 3921	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵))
2		velpw 4559	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	imbi1i 351	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵))
4	3	albii 1838	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵))
5	1, 4	bitri 277	1 ⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1557   ∈ wcel 2141   ⊆ wss 3904  𝒫 cpw 4554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3921  df-pw 4556

This theorem is referenced by:  axpweq  5306  setind2  9700  axgroth5  10779  axgroth6  10783  grumnudlem  44825  ismnuprim  44834