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Theorem incom 3236
Description: Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
incom  |-  ( A  i^i  B )  =  ( B  i^i  A
)

Proof of Theorem incom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ancom 264 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  x  e.  A )
)
2 elin 3227 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3 elin 3227 . . 3  |-  ( x  e.  ( B  i^i  A )  <->  ( x  e.  B  /\  x  e.  A ) )
41, 2, 33bitr4i 211 . 2  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  ( B  i^i  A ) )
54eqriv 2112 1  |-  ( A  i^i  B )  =  ( B  i^i  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1314    e. wcel 1463    i^i cin 3038
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045
This theorem is referenced by:  ineq2  3239  dfss1  3248  in12  3255  in32  3256  in13  3257  in31  3258  inss2  3265  sslin  3270  inss  3274  indif1  3289  indifcom  3290  indir  3293  symdif1  3309  dfrab2  3319  0in  3366  disjr  3380  ssdifin0  3412  difdifdirss  3415  uneqdifeqim  3416  diftpsn3  3629  iunin1  3845  iinin1m  3850  riinm  3853  rintm  3873  inex2  4031  onintexmid  4455  resiun1  4806  dmres  4808  rescom  4812  resima2  4821  xpssres  4822  resindm  4829  resdmdfsn  4830  resopab  4831  imadisj  4869  ndmima  4884  intirr  4893  djudisj  4934  imainrect  4952  dmresv  4965  resdmres  4998  funimaexg  5175  fnresdisj  5201  fnimaeq0  5212  resasplitss  5270  f0rn0  5285  fvun2  5454  ressnop0  5567  fvsnun1  5583  fsnunfv  5587  offres  5999  smores3  6156  phplem2  6713  unfiin  6780  xpfi  6784  endjusym  6947  djucomen  7036  fzpreddisj  9802  fseq1p1m1  9825  hashunlem  10501  zfz1isolem1  10534  znnen  11817  setsfun  11900  setsfun0  11901  setsslid  11915  restin  12251  metreslem  12455  bdinex2  12932
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