Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  if0ab Unicode version

Theorem if0ab 14527
Description: Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition.

Remark: a consequence which could be formalized is the inclusion  |-  if (
ph ,  A ,  (/) )  C_  A and therefore, using elpwg 3583,  |-  ( A  e.  V  ->  if ( ph ,  A ,  (/) )  e.  ~P A
), from which fmelpw1o 14528 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

Assertion
Ref Expression
if0ab  |-  if (
ph ,  A ,  (/) )  =  { x  e.  A  |  ph }
Distinct variable groups:    x, A    ph, x

Proof of Theorem if0ab
StepHypRef Expression
1 noel 3426 . . . . . 6  |-  -.  x  e.  (/)
21intnanr 930 . . . . 5  |-  -.  (
x  e.  (/)  /\  -.  ph )
32biorfi 746 . . . 4  |-  ( ( x  e.  A  /\  ph )  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  (/)  /\  -.  ph ) ) )
43bicomi 132 . . 3  |-  ( ( ( x  e.  A  /\  ph )  \/  (
x  e.  (/)  /\  -.  ph ) )  <->  ( x  e.  A  /\  ph )
)
54abbii 2293 . 2  |-  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  (/)  /\  -.  ph ) ) }  =  { x  |  (
x  e.  A  /\  ph ) }
6 df-if 3535 . 2  |-  if (
ph ,  A ,  (/) )  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  (/)  /\  -.  ph ) ) }
7 df-rab 2464 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
85, 6, 73eqtr4i 2208 1  |-  if (
ph ,  A ,  (/) )  =  { x  e.  A  |  ph }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    \/ wo 708    = wceq 1353    e. wcel 2148   {cab 2163   {crab 2459   (/)c0 3422   ifcif 3534
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-nul 3423  df-if 3535
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator