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Theorem uncom 3130
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 3127 . . 3 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵𝑥𝐴))
31, 2bitr4i 185 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐵𝐴))
43uneqri 3128 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wo 662   = wceq 1287  wcel 1436  cun 2984
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-un 2990
This theorem is referenced by:  equncom  3131  uneq2  3134  un12  3144  un23  3145  ssun2  3150  unss2  3157  ssequn2  3159  undir  3235  dif32  3248  undif2ss  3343  uneqdifeqim  3352  prcom  3495  tpass  3515  prprc1  3527  difsnss  3560  suc0  4205  fvun2  5319  fmptpr  5434  fvsnun2  5440  fsnunfv  5442  omv2  6161  phplem2  6502  undifdc  6564  fzsuc2  9400  fseq1p1m1  9415
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