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

Theorem ifssun 3519
Description: A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
Assertion
Ref Expression
ifssun  |-  if (
ph ,  A ,  B )  C_  ( A  u.  B )

Proof of Theorem ifssun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfif6 3507 . 2  |-  if (
ph ,  A ,  B )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  B  |  -.  ph } )
2 ssrab2 3213 . . 3  |-  { x  e.  A  |  ph }  C_  A
3 ssrab2 3213 . . 3  |-  { x  e.  B  |  -.  ph }  C_  B
4 unss12 3279 . . 3  |-  ( ( { x  e.  A  |  ph }  C_  A  /\  { x  e.  B  |  -.  ph }  C_  B )  ->  ( { x  e.  A  |  ph }  u.  {
x  e.  B  |  -.  ph } )  C_  ( A  u.  B
) )
52, 3, 4mp2an 423 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  -.  ph } )  C_  ( A  u.  B )
61, 5eqsstri 3160 1  |-  if (
ph ,  A ,  B )  C_  ( A  u.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3   {crab 2439    u. cun 3100    C_ wss 3102   ifcif 3505
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-if 3506
This theorem is referenced by:  ifidss  3520  ifelpwung  4442
  Copyright terms: Public domain W3C validator