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Theorem uncom 3062
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 634 . . 3 ((x A x B) ↔ (x B x A))
2 elun 3059 . . 3 (x (BA) ↔ (x B x A))
31, 2bitr4i 176 . 2 ((x A x B) ↔ x (BA))
43uneqri 3060 1 (AB) = (BA)
Colors of variables: wff set class
Syntax hints:   wo 616   = wceq 1228   wcel 1374  cun 2890
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-un 2897
This theorem is referenced by:  equncom  3063  uneq2  3066  un12  3076  un23  3077  ssun2  3082  unss2  3089  ssequn2  3091  undir  3162  dif32  3175  disjpss  3253  undif2ss  3274  uneqdifeqim  3283  prcom  3418  tpass  3438  prprc1  3450  difsnss  3482  suc0  4095  fvun2  5163  fmptpr  5278  fvsnun2  5284  fsnunfv  5286  omv2  5958
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