Theorem ifelpwung 4602

Description: Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)

Assertion

Ref	Expression
ifelpwung	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))

Proof of Theorem ifelpwung

Step	Hyp	Ref	Expression
1		ifssun 3637	. 2 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
2		unexg 4564	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
3		elpw2g 4268	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)))
4	2, 3	syl 14	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)))
5	1, 4	mpbiri 168	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2203  Vcvv 2813   ∪ cun 3209   ⊆ wss 3211  ifcif 3620  𝒫 cpw 3669

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915

This theorem is referenced by:  ifelpwund  4603  ifelpwun  4604