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Theorem uncom 3117
Description: Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
uncom (𝐴𝐵) = (𝐵𝐴)

Proof of Theorem uncom
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 orcom 680 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
2 elun 3114 . . 3 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵𝑥𝐴))
31, 2bitr4i 185 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐵𝐴))
43uneqri 3115 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wo 662   = wceq 1285  wcel 1434  cun 2972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978
This theorem is referenced by:  equncom  3118  uneq2  3121  un12  3131  un23  3132  ssun2  3137  unss2  3144  ssequn2  3146  undir  3215  dif32  3228  undif2ss  3320  uneqdifeqim  3329  prcom  3470  tpass  3490  prprc1  3502  difsnss  3533  suc0  4168  fvun2  5266  fmptpr  5381  fvsnun2  5387  fsnunfv  5389  omv2  6103  phplem2  6378  undiffi  6433  fzsuc2  9161  fseq1p1m1  9176
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