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Theorem uncom 3220
 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 717 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
2 elun 3217 . . 3 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵𝑥𝐴))
31, 2bitr4i 186 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐵𝐴))
43uneqri 3218 1 (𝐴𝐵) = (𝐵𝐴)
 Colors of variables: wff set class Syntax hints:   ∨ wo 697   = wceq 1331   ∈ wcel 1480   ∪ cun 3069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075 This theorem is referenced by:  equncom  3221  uneq2  3224  un12  3234  un23  3235  ssun2  3240  unss2  3247  ssequn2  3249  undir  3326  dif32  3339  undif2ss  3438  uneqdifeqim  3448  prcom  3599  tpass  3619  prprc1  3631  difsnss  3666  suc0  4333  fvun2  5488  fmptpr  5612  fvsnun2  5618  fsnunfv  5621  omv2  6361  phplem2  6747  undifdc  6812  endjusym  6981  fzsuc2  9873  fseq1p1m1  9888  ennnfonelem1  11933  setsslid  12025  exmid1stab  13302
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