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

Theorem rabswap 2889
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 439 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  x  e.  A )
)
21abbii 2550 . 2  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { x  |  (
x  e.  B  /\  x  e.  A ) }
3 df-rab 2716 . 2  |-  { x  e.  A  |  x  e.  B }  =  {
x  |  ( x  e.  A  /\  x  e.  B ) }
4 df-rab 2716 . 2  |-  { x  e.  B  |  x  e.  A }  =  {
x  |  ( x  e.  B  /\  x  e.  A ) }
52, 3, 43eqtr4i 2468 1  |-  { x  e.  A  |  x  e.  B }  =  {
x  e.  B  |  x  e.  A }
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   {crab 2711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-rab 2716
  Copyright terms: Public domain W3C validator