Theorem prsspw 4776

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 4754	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))
4	1, 2, 3	mp2an 698	1 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883  𝒫 cpw 4529  {cpr 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-un 3888  df-ss 3900  df-pw 4531  df-sn 4556  df-pr 4558

This theorem is referenced by:  altxpsspw  36205