Theorem prelpw 5409

Description: An unordered pair of two sets is a member of the powerclass of a class if and only if the two sets are members of that class. (Contributed by AV, 8-Jan-2020.)

Assertion

Ref	Expression
prelpw	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))

Proof of Theorem prelpw

Step	Hyp	Ref	Expression
1		prssg 4786	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
2		prex 5395	. . 3 ⊢ {𝐴, 𝐵} ∈ V
3	2	elpw 4570	. 2 ⊢ ({𝐴, 𝐵} ∈ 𝒫 𝐶 ↔ {𝐴, 𝐵} ⊆ 𝐶)
4	1, 3	bitr4di 289	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ⊆ wss 3917  𝒫 cpw 4566  {cpr 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-pw 4568  df-sn 4593  df-pr 4595

This theorem is referenced by:  prelpwi  5410  hashle2prv  14450  umgrpredgv  29074  prprelb  47521