Theorem if0elpw 4273

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 4604. (Contributed by BJ, 5-May-2026.)

Assertion

Ref	Expression
if0elpw	|- ( A e. V -> if ( ph , A , (/) ) e. ~P A )

Proof of Theorem if0elpw

Step	Hyp	Ref	Expression
1		if0ss 3626	. 2 |- if ( ph , A , (/) ) C_ A
2		elpw2g 4270	. 2 |- ( A e. V -> ( if ( ph , A , (/) ) e. ~P A <-> if ( ph , A , (/) ) C_ A ) )
3	1, 2	mpbiri 168	1 |- ( A e. V -> if ( ph , A , (/) ) e. ~P A )

Syntax hints:   -> wi 4   e. wcel 2205   C_ wss 3213   (/)c0 3510   ifcif 3622   ~Pcpw 3671

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 2216  ax-sep 4230

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673

This theorem is referenced by:  fmelpw1o  7559