Theorem uncom 3408

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	⊢ (A ∪ B) = (B ∪ A)

Proof of Theorem uncom

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orcom 376	. . 3 ⊢ ((x ∈ A ∨ x ∈ B) ↔ (x ∈ B ∨ x ∈ A))
2		elun 3220	. . 3 ⊢ (x ∈ (B ∪ A) ↔ (x ∈ B ∨ x ∈ A))
3	1, 2	bitr4i 243	. 2 ⊢ ((x ∈ A ∨ x ∈ B) ↔ x ∈ (B ∪ A))
4	3	uneqri 3406	1 ⊢ (A ∪ B) = (B ∪ A)

Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ∪ cun 3207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214

This theorem is referenced by:  equncom  3409  uneq2  3412  un12  3421  un23  3422  ssun2  3427  unss2  3434  ssequn2  3436  undir  3504  unineq  3505  dif32  3517  symdifcom  3542  disjpss  3601  undif1  3625  undif2  3626  difcom  3634  uneqdifeq  3638  dfif4  3673  dfif5  3674  prcom  3798  tpass  3818  difsnid  3854  ssunsn2  3865  sspr  3869  sstp  3870  pwadjoin  4119  addccom  4406  sfinltfin  4535  phialllem2  4617  fvun2  5380  fvsnun2  5448  sbthlem1  6203