Theorem ifelpwung 4578

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 3620	. 2 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
2		unexg 4540	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
3		elpw2g 4246	. . 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 2202  Vcvv 2802   ∪ cun 3198   ⊆ wss 3200  ifcif 3605  𝒫 cpw 3652

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894

This theorem is referenced by:  ifelpwund  4579  ifelpwun  4580