Theorem prsspw 4849

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 4827	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))
4	1, 2, 3	mp2an 692	1 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2105  Vcvv 3477   ⊆ wss 3962  𝒫 cpw 4604  {cpr 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-un 3967  df-ss 3979  df-pw 4606  df-sn 4631  df-pr 4633

This theorem is referenced by:  altxpsspw  35958