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

Theorem ifelpwung 4466
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 3540 . 2  |-  if (
ph ,  A ,  B )  C_  ( A  u.  B )
2 unexg 4428 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  e.  _V )
3 elpw2g 4142 . . 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 167 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 103    <-> wb 104    e. wcel 2141   _Vcvv 2730    u. cun 3119    C_ wss 3121   ifcif 3526   ~Pcpw 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797
This theorem is referenced by:  ifelpwund  4467  ifelpwun  4468
  Copyright terms: Public domain W3C validator