Theorem ifelpwung 4625

Description: Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)

Assertion

Ref	Expression
ifelpwung	|- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. ~P ( A u. B ) )

Proof of Theorem ifelpwung

Step	Hyp	Ref	Expression
1		ifssun 3655	. 2 |- if ( ph , A , B ) C_ ( A u. B )
2		unexg 4587	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V )
3		elpw2g 4290	. . 3 |- ( ( A u. B ) e. _V -> ( if ( ph , A , B ) e. ~P ( A u. B ) <-> if ( ph , A , B ) C_ ( A u. B ) ) )
4	2, 3	syl 14	. 2 |- ( ( A e. V /\ B e. W ) -> ( if ( ph , A , B ) e. ~P ( A u. B ) <-> if ( ph , A , B ) C_ ( A u. B ) ) )
5	1, 4	mpbiri 168	1 |- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. ~P ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   _Vcvv 2821   u. cun 3218   C_ wss 3220   ifcif 3638   ~Pcpw 3688

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 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-14 2212  ax-ext 2220  ax-sep 4247  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-uni 3934

This theorem is referenced by:  ifelpwund  4626  ifelpwun  4627