Theorem prsspw 3791

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 3774	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝐶)
4	1	elpw 3607	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 ↔ 𝐴 ⊆ 𝐶)
5	2	elpw 3607	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 ↔ 𝐵 ⊆ 𝐶)
6	4, 5	anbi12i 460	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
7	3, 6	bitr3i 186	1 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3153  𝒫 cpw 3601  {cpr 3619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625

This theorem is referenced by: (None)