Theorem if0elpw 4250

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 4580. (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 3608	. 2 ⊢ if(𝜑, 𝐴, ∅) ⊆ 𝐴
2		elpw2g 4247	. 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 2201   ⊆ wss 3199  ∅c0 3493  ifcif 3604  𝒫 cpw 3653

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 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-ext 2212  ax-sep 4208

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rab 2518  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655

This theorem is referenced by:  fmelpw1o  7470