Theorem prsspw 4778

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)

Hypotheses

Ref	Expression
prsspw.1	⊢ 𝐴 ∈ V
prsspw.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
prsspw	⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))

Proof of Theorem prsspw

Step	Hyp	Ref	Expression
1		prsspw.1	. 2 ⊢ 𝐴 ∈ V
2		prsspw.2	. 2 ⊢ 𝐵 ∈ V
3		prsspwg 4758	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))
4	1, 2, 3	mp2an 689	1 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  Vcvv 3431   ⊆ wss 3888  𝒫 cpw 4535  {cpr 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3433  df-un 3893  df-in 3895  df-ss 3905  df-pw 4537  df-sn 4564  df-pr 4566

This theorem is referenced by:  altxpsspw  34266