Theorem ifelpwung 4576

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 3618	. 2 ⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
2		unexg 4538	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
3		elpw2g 4244	. . 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 2200  Vcvv 2800   ∪ cun 3196   ⊆ wss 3198  ifcif 3603  𝒫 cpw 3650

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892

This theorem is referenced by:  ifelpwund  4577  ifelpwun  4578