Theorem if0elpw 4290

Description: A conditional class with the False alternative being sent to the empty class is an element of the powerset of the class corresponding to the True alternative when that class is a set. This statement requires fewer axioms than the general case ifelpwung 4622. (Contributed by BJ, 5-May-2026.)

Assertion

Ref	Expression
if0elpw	⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴)

Proof of Theorem if0elpw

Step	Hyp	Ref	Expression
1		if0ss 3639	. 2 ⊢ if(𝜑, 𝐴, ∅) ⊆ 𝐴
2		elpw2g 4287	. 2 ⊢ (𝐴 ∈ 𝑉 → (if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴 ↔ if(𝜑, 𝐴, ∅) ⊆ 𝐴))
3	1, 2	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2209   ⊆ wss 3220  ∅c0 3520  ifcif 3635  𝒫 cpw 3685

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687

This theorem is referenced by:  fmelpw1o  7596