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Theorem incom 3365
Description: Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
incom (𝐴𝐵) = (𝐵𝐴)

Proof of Theorem incom
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ancom 266 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
2 elin 3356 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 elin 3356 . . 3 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵𝑥𝐴))
41, 2, 33bitr4i 212 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥 ∈ (𝐵𝐴))
54eqriv 2202 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1373  wcel 2176  cin 3165
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172
This theorem is referenced by:  ineq2  3368  dfss1  3377  in12  3384  in32  3385  in13  3386  in31  3387  inss2  3394  sslin  3399  inss  3403  indif1  3418  indifcom  3419  indir  3422  symdif1  3438  dfrab2  3448  0in  3496  disjr  3510  ssdifin0  3542  difdifdirss  3545  uneqdifeqim  3546  diftpsn3  3774  iunin1  3992  iinin1m  3997  riinm  4000  rintm  4020  inex2  4179  onintexmid  4621  resiun1  4978  dmres  4980  rescom  4984  resima2  4993  xpssres  4994  resindm  5001  resdmdfsn  5002  resopab  5003  imadisj  5044  ndmima  5059  intirr  5069  djudisj  5110  imainrect  5128  dmresv  5141  resdmres  5174  funimaexg  5358  fnresdisj  5386  fnimaeq0  5397  resasplitss  5455  f0rn0  5470  fvun2  5646  ressnop0  5765  fvsnun1  5781  fsnunfv  5785  offres  6220  smores3  6379  phplem2  6950  unfiin  7023  xpfi  7029  endjusym  7198  djucomen  7328  fzpreddisj  10193  fseq1p1m1  10216  hashunlem  10949  zfz1isolem1  10985  fprodsplit  11908  znnen  12769  setsfun  12867  setsfun0  12868  setsslid  12883  ressressg  12907  restin  14648  metreslem  14852  perfectlem2  15472  bdinex2  15836
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