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Theorem uncom 3279
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 728 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
2 elun 3276 . . 3 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵𝑥𝐴))
31, 2bitr4i 187 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐵𝐴))
43uneqri 3277 1 (𝐴𝐵) = (𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wo 708   = wceq 1353  wcel 2148  cun 3127
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133
This theorem is referenced by:  equncom  3280  uneq2  3283  un12  3293  un23  3294  ssun2  3299  unss2  3306  ssequn2  3308  undir  3385  dif32  3398  undif2ss  3498  uneqdifeqim  3508  prcom  3667  tpass  3687  prprc1  3699  difsnss  3737  suc0  4407  fvun2  5578  fmptpr  5703  fvsnun2  5709  fsnunfv  5712  omv2  6459  phplem2  6846  undifdc  6916  endjusym  7088  fzsuc2  10052  fseq1p1m1  10067  ennnfonelem1  12378  setsslid  12482  exmid1stab  14372
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