ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifelpwung Unicode version

Theorem ifelpwung 4510
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
StepHypRef Expression
1 ifssun 3571 . 2  |-  if (
ph ,  A ,  B )  C_  ( A  u.  B )
2 unexg 4472 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  e.  _V )
3 elpw2g 4185 . . 3  |-  ( ( A  u.  B )  e.  _V  ->  ( if ( ph ,  A ,  B )  e.  ~P ( A  u.  B
)  <->  if ( ph ,  A ,  B )  C_  ( A  u.  B
) ) )
42, 3syl 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 )
) )
51, 4mpbiri 168 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  ~P ( A  u.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2164   _Vcvv 2760    u. cun 3151    C_ wss 3153   ifcif 3557   ~Pcpw 3601
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pr 4238  ax-un 4462
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836
This theorem is referenced by:  ifelpwund  4511  ifelpwun  4512
  Copyright terms: Public domain W3C validator