Theorem prsspw 3752

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} ⊆ 𝒫 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 3736	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝐶)
4	1	elpw 3572	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 ↔ 𝐴 ⊆ 𝐶)
5	2	elpw 3572	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 ↔ 𝐵 ⊆ 𝐶)
6	4, 5	anbi12i 457	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
7	3, 6	bitr3i 185	1 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  𝒫 cpw 3566  {cpr 3584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590

This theorem is referenced by: (None)