Theorem prelpw 5437

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 4815	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
2		prex 5423	. . 3 ⊢ {𝐴, 𝐵} ∈ V
3	2	elpw 4599	. 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 205   ∧ wa 395   ∈ wcel 2098   ⊆ wss 3941  𝒫 cpw 4595  {cpr 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-pw 4597  df-sn 4622  df-pr 4624

This theorem is referenced by:  prelpwi  5438  hashle2prv  14441  umgrpredgv  28894  prprelb  46730