Theorem if0elpw 4270

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 4601. (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 3623	. 2 ⊢ if(𝜑, 𝐴, ∅) ⊆ 𝐴
2		elpw2g 4267	. 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 2203   ⊆ wss 3210  ∅c0 3507  ifcif 3619  𝒫 cpw 3668

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 619  ax-in2 620  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-ext 2214  ax-sep 4227

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-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670

This theorem is referenced by:  fmelpw1o  7556