ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  incom GIF version

Theorem incom 3129
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 253 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
2 elin 3126 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 elin 3126 . . 3 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵𝑥𝐴))
41, 2, 33bitr4i 201 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥 ∈ (𝐵𝐴))
54eqriv 2037 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wa 97   = wceq 1243  wcel 1393  cin 2916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924
This theorem is referenced by:  ineq2  3132  dfss1  3141  in12  3148  in32  3149  in13  3150  in31  3151  inss2  3158  sslin  3163  inss  3166  indif1  3182  indifcom  3183  indir  3186  symdif1  3202  dfrab2  3212  disjr  3269  ssdifin0  3304  difdifdirss  3307  uneqdifeqim  3308  diftpsn3  3505  iunin1  3721  iinin1m  3726  riinm  3729  rintm  3744  inex2  3892  onintexmid  4297  resiun1  4630  dmres  4632  rescom  4636  resima2  4644  xpssres  4645  resopab  4652  imadisj  4687  ndmima  4702  intirr  4711  djudisj  4750  imainrect  4766  dmresv  4779  resdmres  4812  funimaexg  4983  fnresdisj  5009  fnimaeq0  5020  resasplitss  5069  fvun2  5240  ressnop0  5344  fvsnun1  5360  fsnunfv  5363  offres  5762  smores3  5908  phplem2  6316  fzpreddisj  8933  fseq1p1m1  8956  bdinex2  10020
  Copyright terms: Public domain W3C validator