MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabswap Unicode version

Theorem rabswap 2719
Description: Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
Assertion
Ref Expression
rabswap  |-  { x  e.  A  |  x  e.  B }  =  {
x  e.  B  |  x  e.  A }

Proof of Theorem rabswap
StepHypRef Expression
1 ancom 437 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  x  e.  A )
)
21abbii 2395 . 2  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { x  |  (
x  e.  B  /\  x  e.  A ) }
3 df-rab 2552 . 2  |-  { x  e.  A  |  x  e.  B }  =  {
x  |  ( x  e.  A  /\  x  e.  B ) }
4 df-rab 2552 . 2  |-  { x  e.  B  |  x  e.  A }  =  {
x  |  ( x  e.  B  /\  x  e.  A ) }
52, 3, 43eqtr4i 2313 1  |-  { x  e.  A  |  x  e.  B }  =  {
x  e.  B  |  x  e.  A }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-rab 2552
  Copyright terms: Public domain W3C validator