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Theorem uncom 3087
 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 647 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
2 elun 3084 . . 3 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵𝑥𝐴))
31, 2bitr4i 176 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐵𝐴))
43uneqri 3085 1 (𝐴𝐵) = (𝐵𝐴)
 Colors of variables: wff set class Syntax hints:   ∨ wo 629   = wceq 1243   ∈ wcel 1393   ∪ cun 2915 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922 This theorem is referenced by:  equncom  3088  uneq2  3091  un12  3101  un23  3102  ssun2  3107  unss2  3114  ssequn2  3116  undir  3187  dif32  3200  disjpss  3278  undif2ss  3299  uneqdifeqim  3308  prcom  3446  tpass  3466  prprc1  3478  difsnss  3510  suc0  4148  fvun2  5240  fmptpr  5355  fvsnun2  5361  fsnunfv  5363  omv2  6045  phplem2  6316  fzsuc2  8941  fseq1p1m1  8956
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