Theorem prsspw 3878

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.)

Hypotheses

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

Assertion

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

Proof of Theorem prsspw

Step	Hyp	Ref	Expression
1		prsspw.1	. . 3 ⊢ A ∈ V
2		prsspw.2	. . 3 ⊢ B ∈ V
3	1, 2	prss 3861	. 2 ⊢ ((A ∈ ℘C ∧ B ∈ ℘C) ↔ {A, B} ⊆ ℘C)
4	1	elpw 3728	. . 3 ⊢ (A ∈ ℘C ↔ A ⊆ C)
5	2	elpw 3728	. . 3 ⊢ (B ∈ ℘C ↔ B ⊆ C)
6	4, 5	anbi12i 678	. 2 ⊢ ((A ∈ ℘C ∧ B ∈ ℘C) ↔ (A ⊆ C ∧ B ⊆ C))
7	3, 6	bitr3i 242	1 ⊢ ({A, B} ⊆ ℘C ↔ (A ⊆ C ∧ B ⊆ C))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  ℘cpw 3722  {cpr 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259  df-pw 3724  df-sn 3741  df-pr 3742

This theorem is referenced by: (None)