Theorem prelpw 4270

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 3796	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
2		prexg 4266	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
3		elpwg 3629	. . 3 ⊢ ({𝐴, 𝐵} ∈ V → ({𝐴, 𝐵} ∈ 𝒫 𝐶 ↔ {𝐴, 𝐵} ⊆ 𝐶))
4	2, 3	syl 14	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ∈ 𝒫 𝐶 ↔ {𝐴, 𝐵} ⊆ 𝐶))
5	1, 4	bitr4d 191	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2177  Vcvv 2773   ⊆ wss 3170  𝒫 cpw 3621  {cpr 3639

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645

This theorem is referenced by:  umgrpredgv  15821