Theorem prelpw 5394

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 4775	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
2		prex 5382	. . 3 ⊢ {𝐴, 𝐵} ∈ V
3	2	elpw 4558	. 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 2113   ⊆ wss 3901  𝒫 cpw 4554  {cpr 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-pw 4556  df-sn 4581  df-pr 4583

This theorem is referenced by:  prelpwi  5395  hashle2prv  14401  umgrpredgv  29213  prprelb  47758