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Theorem incom 3342
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 266 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  x  e.  A )
)
2 elin 3333 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3 elin 3333 . . 3  |-  ( x  e.  ( B  i^i  A )  <->  ( x  e.  B  /\  x  e.  A ) )
41, 2, 33bitr4i 212 . 2  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  ( B  i^i  A ) )
54eqriv 2186 1  |-  ( A  i^i  B )  =  ( B  i^i  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364    e. wcel 2160    i^i cin 3143
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150
This theorem is referenced by:  ineq2  3345  dfss1  3354  in12  3361  in32  3362  in13  3363  in31  3364  inss2  3371  sslin  3376  inss  3380  indif1  3395  indifcom  3396  indir  3399  symdif1  3415  dfrab2  3425  0in  3473  disjr  3487  ssdifin0  3519  difdifdirss  3522  uneqdifeqim  3523  diftpsn3  3748  iunin1  3966  iinin1m  3971  riinm  3974  rintm  3994  inex2  4153  onintexmid  4587  resiun1  4941  dmres  4943  rescom  4947  resima2  4956  xpssres  4957  resindm  4964  resdmdfsn  4965  resopab  4966  imadisj  5005  ndmima  5020  intirr  5030  djudisj  5071  imainrect  5089  dmresv  5102  resdmres  5135  funimaexg  5316  fnresdisj  5342  fnimaeq0  5353  resasplitss  5411  f0rn0  5426  fvun2  5600  ressnop0  5714  fvsnun1  5730  fsnunfv  5734  offres  6155  smores3  6313  phplem2  6876  unfiin  6949  xpfi  6953  endjusym  7120  djucomen  7240  fzpreddisj  10096  fseq1p1m1  10119  hashunlem  10811  zfz1isolem1  10847  fprodsplit  11632  znnen  12444  setsfun  12542  setsfun0  12543  setsslid  12558  ressressg  12580  restin  14113  metreslem  14317  bdinex2  15089
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